magnetic phases in the hubbard model

Magnetic Phases in theHubbard Model

Dissertation

zur Erlangung des mathematisch-naturwissenschaftlichen

Doktorgrades

”Doctor rerum naturalium”

der Georg-August-Universitat Gottingen

vorgelegt von

Robert Peters

aus Karl-Marx-Stadt (jetzt Chemnitz)

Gottingen, 2009

D 7

Referent: Prof. Dr. Thomas Pruschke

Koreferent: Prof. Dr. Frithjof Anders

Tag der mundlichen Prufung : 19.11.2009

4

Lines from “Hsin-hsin Ming”

Shinjinmei (or Shinjin no Mei) (Japanese)

By Third Ch’an Patriarch Chien-chih Seng-ts’an

“Kon ni ki sureba shi o e. Sho ni shitagaeba shu o shissu.”

Contents

1 Magnetism and Strong Correlations 7

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Transition Metal Oxides . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Light Transition Metal Oxides . . . . . . . . . . . . . . . 111.2.2 Cobaltates . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2.3 Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2.4 Manganites . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.4 Exchange Interactions . . . . . . . . . . . . . . . . . . . . . . . . 201.5 Types of Magnetic Order . . . . . . . . . . . . . . . . . . . . . . 221.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Dynamical Mean Field Theory 25

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Cavity Construction . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Dynamical Mean Field Theory . . . . . . . . . . . . . . . . . . . 282.4 Magnetic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.1 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . 302.4.2 Antiferromagnetism . . . . . . . . . . . . . . . . . . . . . 32

2.5 Bethe lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5.1 Bethe lattice with next nearest neighbor hopping . . . . . 33

3 Impurity Solver - Theoretical Background 39

3.1 The Anderson Impurity Model . . . . . . . . . . . . . . . . . . . 393.2 Discretization of the Band States . . . . . . . . . . . . . . . . . . 413.3 Numerical Renormalization Group . . . . . . . . . . . . . . . . . 42

3.3.1 Discretization within the NRG . . . . . . . . . . . . . . . 423.3.2 Iterative diagonalization . . . . . . . . . . . . . . . . . . . 433.3.3 Calculation of impurity properties . . . . . . . . . . . . . 45

6 Contents

3.3.4 Calculating the Self-energy . . . . . . . . . . . . . . . . . 483.3.5 Two-orbital Anderson model . . . . . . . . . . . . . . . . 49

3.4 Density Matrix Renormalization Group . . . . . . . . . . . . . . 503.4.1 Iterative diagonalization . . . . . . . . . . . . . . . . . . . 503.4.2 Calculation of dynamical properties . . . . . . . . . . . . 523.4.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 NRG and DMRG Spectral Functions for the Impurity Model 55

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2 Used Discretization Schemes . . . . . . . . . . . . . . . . . . . . . 574.3 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.1 Matrix Inversion . . . . . . . . . . . . . . . . . . . . . . . 584.3.2 Maximum Entropy Ansatz . . . . . . . . . . . . . . . . . . 59

4.4 Comparison between DMRG and NRG . . . . . . . . . . . . . . . 604.4.1 Single Impurity Anderson Model . . . . . . . . . . . . . . 604.4.2 Two-Orbital Anderson Model . . . . . . . . . . . . . . . . 64

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5 One-orbital Hubbard model 67

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 Semi-elliptic Density of States . . . . . . . . . . . . . . . . . . . . 68

5.2.1 Metal Insulator Transition . . . . . . . . . . . . . . . . . . 685.2.2 Magnetic Phase Diagram . . . . . . . . . . . . . . . . . . 695.2.3 Spin Density Waves . . . . . . . . . . . . . . . . . . . . . 72

5.3 Comparison of DMRG and NRG . . . . . . . . . . . . . . . . . . 775.4 Frustrated Bethe Lattice . . . . . . . . . . . . . . . . . . . . . . . 80

5.4.1 Half-Filling . . . . . . . . . . . . . . . . . . . . . . . . . . 805.4.2 Antiferromagnetism at half-filling . . . . . . . . . . . . . . 825.4.3 Nearly fully frustrated system . . . . . . . . . . . . . . . . 855.4.4 Doped System . . . . . . . . . . . . . . . . . . . . . . . . 895.4.5 Ferromagnetism in the Frustrated system . . . . . . . . . 925.4.6 Energies for the different magnetic states . . . . . . . . . 95

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6 Two-Orbital Hubbard Model 99

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.2 Two Site Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.3 Magnetic Phase Diagram . . . . . . . . . . . . . . . . . . . . . . 1036.4 Quarter-Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.4.1 Ferromagnetic Metal Insulator Transition . . . . . . . . . 1076.4.2 Antiferromagnetism and Charge Order . . . . . . . . . . . 113

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7 Summary and Outlook 117

Appendices 121

A Calculation of the Ground State Energy 123

CHAPTER 1

Magnetism and Strong Correlations

1.1 Introduction

The ability of some materials to attract others has fascinated many people forover 2500 years. Perhaps the widest known legend is by Pliny the elder, whodescribed how the shepherd “Magnes” stuck with his iron nails in his shoes toa black stone while he pastured his flocks. Besides this legend the first definitestatements were by Thales of Miletus about 585 B.C., saying that lodestone, thenaturally occurring mineral magnetite Fe3O4, can attract iron [1]. The namemagnetite most probably comes from the old Greek city “Magnesia” in AsiaMinor, where large depositions of the magnetic mineral could be found [1, 2].Since then there have been developed many explanations and applications ofthe phenomenon magnetism. The first widely known application is the com-pass, which was most likely invented in China and was then used since the 11thcentury in Europe [1, 2]. William Gilbert, who was an English physicist andnatural scientist, wrote in 1600 “De Magnete” [3], in which he describes exper-iments and phenomenology of magnets and also states that the earth itself isa big magnet explaining the compass. This was the first modern attempt tounderstand magnetism. In the 19th century a big breakthrough was achievedby many scientists1, who noticed and explained the connection between elec-tricity and magnetism [4]. But still the origin of remanent magnetism couldnot be explained. It was not before the development of the quantum theory ofmaterials [5] and the discovery of the spin in the first half of the 20th centurythat the way was free for explaining and describing the source of magnetism.The application of magnetic fields is today indispensable. What began with the

1to mention some of them: C Oersted, J Biot, F Savart, A Ampere, M Faraday and J

Maxwell

8 Magnetism and Strong Correlations

compass, is nowadays used in power production, means of transportation andcomputer devices like storage and memory. Today’s magnetic materials used inelectronics are not some stones found in nature, but precisely engineered thinfilms and multilayer structures on the nanometer scale [6]. The guiding princi-ples today are “smaller and faster”. Thus the exact study and understanding ofthe underlying physics in the nowadays used materials and structures is crucialfor designing new devices.Today’s theories about the origin of magnetism rely on the concepts of charge,spin and quantum mechanics developed in the early 20th century [5]. Thepioneers of the quantum theory noticed that the angular momentum and thespin of electrons can form magnetic moments, which eventually align. Thisarrangement of moments is the origin of the magnetic fields observed aroundpermanent magnets [1]. The widely known magnetism often actually denotesferromagnetism in which all magnetic moments are aligned parallel. But in gen-eral it is clear that the moments may align in more complicated arrangements,if this is energetically favorable. In this work I will use the term “magnetic or-dering” in the sense of some periodic, long range alignment of spins regardlessof the definite arrangement.The Curie temperature of iron, the temperature below which the magneticmoments are aligned parallel, is 1043K [7]. One should now ask, what is theforce, which is responsible for this order. From a classical electrodynamicslecture [4], one knows that magnetic moments act on each other via a dipoleinteraction. One can simply estimate the strength of this interaction in ironto be 0.1K [7]. For sure, this force is too weak to produce such a high Curietemperature. As it will become clear below, the reasons for the alignment areactually the Coulomb interaction between electrons, and the Pauli exclusionprinciple.If one looks at the periodic table of elements and searches for the elements,which show ferromagnetic behavior at room temperature one will only find Fe,Co, Ni. All three have partially filled d-orbitals. When looking for chemicalcompounds, which order at room temperature or below, one must notice thatoften partially filled d- or f-orbitals are involved. So there are three majorgroups showing magnetic ordering: transition metals, transition metal oxidesand rare earth elements as well as actinides and their compounds [7]. Besidesthese groups one should also mention that some organic compounds includingtransition metal atoms can show magnetic ordering [7].

1.2 Transition Metal Oxides

The above discussion results in the obvious question: How is magnetic orderingconnected to d-orbitals? Consider the 3d-orbital atomic wave function in atransition metal. The wave function of a free atom can always be written asa product of a radial part and a spherical harmonics. The 3d-orbitals haveangular momentum l = 2. As the 1s−, 2s−, 2p−, 3s− and 3p-orbitals havedifferent angular momentum, the 3d-orbitals are orthogonal to them becauseof their spherical harmonics. Thus the radial part of the wave function has

1.2 Transition Metal Oxides 9

Figure 1.1: Cubic perovskite structure of a typical transition metal oxide. The greenspheres are the transition metal atoms (M), which are surrounded by octahedra ofoxygen drawn as small blue spheres. The big red spheres represent the A-atoms ofAMO3.

no node and does not “extend” as far as the 3s- or 3p-orbitals. In contrast,the 4s-orbitals with angular momentum l = 0 extend very far from the nucleusof the atoms. The 4s-orbitals lie energetically lower than the 3d-orbitals andare filled before them. The electrons in the 4s-bands are most important forscreening the Coulomb interaction in the 3d-band [8, 9]. When the energydifference between both bands is large, the screening is very poor leading to astrong Coulomb interaction between the d-electrons. On the other hand, the3d-orbitals extend less and thus have only a small overlap with each other.Therefore the bandwidth is much smaller, resulting in a tendency to localize,which can lead to more pronounced correlation effects.The following elements are most important when studying the physics of 3d-transition metals: Ti, V, Cr, Mn, Fe, Co, Ni and Cu. These elements often formcompounds with oxygen [10]. The typical structure of transition metal oxidesis the cubic perovskite like AMO3, see figure 1.1, where the transition metalatoms (M), green spheres, are surrounded by an octahedra of oxygen atoms (O),blue spheres. The A-atoms, red spheres, represent some heavy elements. Thesecompounds can contain partially filled d-shells, which turn out to be especiallyimportant for explaining their physical properties. The transition metal atomsare situated in an octahedral environment. Crystal field theory predicts inthis case, that the 5-fold degenerate d-orbitals split into 3-fold t2g-levels and2-fold eg-levels [7]. In figure 1.2 one can see the level splitting of d-orbitals in


Ener

gy

eg

t2g

Spherical

Octahedral

Tetragonal

dx2−y2

d3z2−r2

dxy

dxz, dyz

Symmetry groups-

typical t2g - p typical eg - p

xy

3z2 − r2

x2 − y2

Figure 1.2: Energetic splitting of the d-orbitals in different symmetry groups. Theenergy is in arbitrary units, as the picture is only supposed to illustrate the splitting.The exact position of the levels depend on the specific system. Left: 5-fold degeneratelevels in spherical symmetry. Next: Splitting in octahedral symmetry into a t2g- andeg-band. The t2g-levels have lower energy in a cubic perovskite. Next: Splitting intetragonal symmetry, as it occurs in a static Jahn-Teller distortion [11]. Right: Shapeand classification of the d-orbitals. The spread in “z”-direction is color coded. Thepictures at the bottom show typical configurations between one d- and a neighboringp-orbital showing that the overlap is largest for the eg-levels.


octahedral and tetrahedral environment. On the right side of the figure one cansee the basic shape of the d-orbitals. In the cubic perovskite the orbitals of thet2g-wave functions point to the corners of the cube while the eg-wave functionspoint to the faces of the cube where the oxygen atoms are located. In thisconfiguration the t2g-levels are lower in energy due to the repulsive electrostaticinteraction with the occupied p-orbitals of oxygen. The splitting of t2g- andeg-orbitals can be as large as 2−3 eV [10]. In transition metal oxides the directoverlap between the d-orbitals of neighboring sites is often negligible. Thehopping of the electrons from one transition metal atom to another is mediatedby the oxygen surrounding these [10]. The strength of this process depends onthe exact positions and the energy levels of the oxygen and the transition metalatoms, but it can be assumed to be small. From this fact it is clear that theelectrical resistivity of these compounds can show interesting properties. Bandtheory predicts that a partially filled band is always a metal, but it was alreadyreported in 1937 by J. de Boer and E. Verwey [12,13] that many transition metaloxides with partially filled d-orbitals are very poor metals or even insulators [14].In a very simplified picture one can imagine that at low temperatures in a half-filled band one electron sits at each lattice site. If now an electron moves throughthe lattice, it has to pay an extra energy when it visits a site where an electronis already located. For strong enough repulsive interaction, this will prevent theelectrons from moving at all [13]. This is an example of a Mott insulator, whichis due to strong repulsive electron-electron interactions. An insulator formed bya partially filled shell will have localized electrons, which may form magneticmoments due to their spin and angular momentum. As this localization isa collective phenomenon of the electrons there are correlations between themeventually aligning the magnetic moments. This shows the close connectionbetween strong correlations, metal insulator transition and magnetism. In thefollowing I will address some particular interesting examples of transition metaloxides. Some of the phenomena discussed in the next paragraphs will appearagain later in this thesis.

1.2.1 Light Transition Metal Oxides

A prominent example showing a metal insulator transition is Vanadiumoxide,V2O3 [14, 15], for with the phase diagram is depicted in figure 1.3. For lowtemperatures and low pressure an antiferromagnetically ordered state is real-ized. Applying pressure does not change the local electron-electron interactionin this compound, but it changes the overlap between the electronic wave func-tions thus changing the bandwidth. Therefore the ratio between the interactionand the bandwidth is tuned. Strikingly, for moderate temperatures a param-agnetic metal insulator transition, a drop in the resistivity of several orders ofmagnitude, can be observed. The transition can be triggered by changing thepressure, temperature or chemical substitution. Vanadiumoxide crystallizesin the corundum-structure, which introduces frustration to the system [14].Frustration means here that there are competing magnetic interactions, whichcannot all be satisfied simultaneously. This effect can be the reason, why theantiferromagnetic state in V2O3 becomes quickly unstable, when increasing the


4− 4 0

∆Pressure

Antiferromagnetic Insulator100

200

Tem

pera

ture

in K

300

400

Insulator

Critical Point

Metal

p [4 kbar/division]

Figure 1.3: Phase diagram of V2O3 redrawn after D. McWhan et al. [15]. For lowtemperatures the system forms an antiferromagnetic insulator which becomes unstabletowards a paramagnetic metal with increasing pressure. Interestingly, for tempera-tures between 200 K< T < 400 K a pressure dependent paramagnetic metal insulatortransition is observed.

temperature and a paramagnetic metal insulator transition can be observed.Other light transition metal oxides with perovskite structure are, for exam-ple, R1−xAxTiO3 and R1−xAxVO3 (R: trivalent cations, A: divalent cations)with their parent compounds like YTiO3 and LaVO3. In these substances thet2g-bands are filled with one or two electrons respectively, thus being partiallyfilled. Nevertheless both materials are insulators [16, 17]. They are appropri-ate systems for experimental investigation of the paramagnetic metal insulatortransition triggered by doping [18]. For example, with increasing hole con-centration in La1−xSrxTiO3, one can observe a metal insulator transition forx = 0.05.

1.2.2 Cobaltates

Cobaltates have been studied because of their quite unique magnetic and trans-port properties. For example, NaCo2O4 shows a large thermoelectric effect [19].A thermoelectric device creates a voltage, when there is a temperature gradientpresent in the material. The discussed applications are manifold, for exampleutilization of waste heat or power supply for deep space probes [20]. The d-orbitals in Cobaltates are filled with five to six electrons [10, 14]. There is astrong competition between different electron configurations, so-called valence-fluctuations [21]. At low temperatures all electrons occupy the energeticallylower t2g-band forming a low spin configuration. But it is also possible thatone or two electrons occupy the eg-band forming intermediate or high-spinconfigurations. The competition between these states lead to very interesting


Temperature in K0 200 400 600 800

0.005

0.01

0.015χ

inµ

B/C

o

Figure 1.4: Magnetic susceptibility χ versus temperature of LaCoO3 showing the non-magnetic ground state and the transition to the high spin state, indicated by themaximum in the curve. (This is a sketch redrawn after M. Imada et al. Rev. Mod.Phys. 70 1039 (page 1235) [14])

transport and magnetic behavior of LaCoO3 shown in figure 1.4 [14,22,23]. Atlow temperatures this compound is a non-magnetic insulator with low-spin statemeaning the t2g-band is fully occupied. However, for increasing temperature thespin state changes and the magnetic susceptibility shows a maximum at about100K. LaCoO3 and the hole-doped La1−xSrxCoO3 are excellent examples howthe different orbitals can influence the physics.

1.2.3 Cuprates

High-temperature (high-TC) superconductivity was first discovered in 1986 inthe copper oxide La1.85Ba0.15CuO4 [24, 25]. High-TC superconductivity refersto compounds with transition temperatures TC > 30K and is a very activefield in physics. Today a large number of cuprate compounds showing high-TC

superconductivity are known. Their structure is typically a layered perovskitestructure with CuO2 layers [14, 25]. Soon after the discovery it was evidentthat strong electron correlations are important in these compounds [25, 26].La2CuO4 is an antiferromagnetic insulator with a Neel-temperature aroundTN = 300K. The degeneracy of the eg-orbitals is lifted due to the anisotropy,which mainly originates from the two-dimensionality. La2CuO4 has 9 electronsin the d-orbitals leading to a completely filled t2g- and d3z2−r2-shell while theremaining dx2−y2-shell is half-filled. The half-filled band combined with theinsulating behavior is a strong evidence for the importance of strong electron


correlations. The superconductivity arises when a small amount of hole-carriersare introduced by doping for example with Ba [25]. For some compounds thetransition temperature can be higher than T = 77K, the boiling point of nitro-gen [27].

1.2.4 Manganites

The last interesting material class which I want to introduce here, are themanganites [14, 28–31]. They became famous because of the colossal magnetoresistance effect. This is a large change in resistivity upon changing the strengthof the magnetic field near the Curie temperature. If one now considers, thatreading and writing of memory in a computer or other electronic devices useexactly such resistance changes, one can easily imagine, why these materialsreceives so much interest. Manganites are composed of MnO3 octahedra withperovskite structure. The manganese d-orbitals have no direct overlap witheach other. The overlap is mediated by the inbetween lying oxygen atoms.A typical example for a manganite compound is La1−xCaxMnO3. Figure 1.5shows a sample picture and some particular important experiments. All threepictures correspond to La0.75Ca0.25MnO3. The experiments were performed atthe University of Gottingen by S. A. Koster and V. Moshnyaga. The upperleft picture shows a photograph of such a sample, while the upper right pictureshows a high resolution transmission electron microscopy image of this com-pound, which was grown on MgO. In this picture a superstructure of La andCa was found. Both upper pictures are supposed to give the reader an impres-sion how the samples look like, for which the experiments are performed. Thelower pictures show the results of three typical experiments. The resistivity isplotted versus the temperature. For zero magnetic field, the resistivity increasesfor decreasing temperatures until the sample reaches its Curie temperature. Atthe Curie temperature there is a sudden drop in the resistivity. By applying amagnetic field of H = 5T one can increase the Curie temperature. The resis-tivity of the system shows a much smaller initial increase, but now decreasesat higher temperatures. This change is called the colossal magneto resistanceeffect [29], as the resistivity around the Curie temperatures changes several or-der of magnitude. The lower panels show the ferromagnetic magnetization ofthis compound for low temperatures and a ferromagnetic hysteresis curve, whenapplying a magnetic field. Such phase transitions into magnetic phases will bethe principal focus of this work.Additionally, the phase diagram of this compound is shown in figure 1.6. Themanganese d-orbitals are filled with n = 4 − x electrons. The electronic con-figuration is such that strong electron-electron interactions including strongferromagnetic Hund’s coupling play an important role [33, 34]. Three of the4 − x electrons occupy the t2g-orbitals. Ferromagnetic Hund’s coupling forcesthem to align forming a S = 3/2-spin. As the exchange between the t2g-states ofdifferent manganese atoms is negligible, these states can be treated as localizedspins, which couple ferromagnetically to the eg-band. The remaining electronsoccupy the eg-band. There is a hopping of the eg-electrons between differ-ent manganese atoms. Similar as in V2O3, the screening is not very efficient


-10000 -5000 0 5000 10000-750-500-250

0250500750

M/V

[em

u/cm

3 ]

Field [Oe]

Hystereses10K

50 100 150 200 250 300

0.0

0.2

0.4

0.6

0.8

1.0

M [e

mu1

0-3]

T [K]

Magnetization

100 Oe

150 200 250 300

0.00.51.01.52.02.5

R [k

Ω]

T [K]

Resistivity H=0

H=50kOe

Figure 1.5: All three pictures correspond to La0.75Ca0.25MnO3. Top Left: Picture of asample, taken by S. A. Koster at the University of Gottingen. Top Right: High Resolu-tion Transmission Electron Microscopy image of such a sample. The La0.75Ca0.25MnO3

thin film is grown on MgO. Remarkably, V. Moshnyaga et al. found a perovskite su-perstructure due to La/Ca ordering. (Reprinted with permission from V. Moshnyagaet al. Phys. Rev. Lett. 97, 107205 (2006). COPYRIGHT 2006 by American PhysicalSociety [32]) Bottom: The picture shows three experiments, performed by S. A. Kosterand V. Moshnyaga, in the ferromagnetic region of this compound. The upper panelshows the resistance R for two different magnetic fields H . The middle panel showsthe ferromagnetic magnetization M via temperature T and the lower panel shows aferromagnetic hysteresis curve M(H).


Figure 1.6: Magnetic Phase diagram of La1−xCaxMnO3. Redrawn after S. Cheongand H. Hwang “Ferromagnetism vs. Charge/Orbital Ordering” in Colossal magnetoresistive oxides by Y. Tokura [29].

leading to strong Coulomb interaction in the eg-band. For understanding thephysics of manganites, the Jahn-Teller distortion is crucial [35, 36]. The sym-metry reduction leads to the situation that the eg-band is further split. Thisdistortion especially occurs, if there is one eg-electron at one site, correspond-ing to quarter-filling. The low temperature magnetic phase diagram is shownin figure 1.6 [29]. LaMnO3 is known to be a A-type antiferromagnet, meaninga ferromagnetic coupling in the orthorhombic planes and an antiferromagneticcoupling perpendicular. For small x of Ca doping, the Jahn-Teller distortionis very strong, splitting the energy of the eg-orbitals, and the system forms acanted antiferromagnetic state (CAF), which is orbitally ordered. For largerdoping the system undergoes a transition to a ferromagnetic insulator (FI) andlater for 0.2 ≤ x ≤ 0.5 to a ferromagnetic metal (FM). This is the regime,where the CMR effect can be observed, as the paramagnetic phase above theCurie temperature is an insulator. For x > 0.5 there are less than 0.5 electronsper site in the eg-band, and the system forms an antiferromagnetic state again(AF), where the electrons are localized in stripes. Such a charge ordering isdenoted with (CO) in the phase diagram. At the end of the La1−xCaxMnO3

series is CaMnO3, which is like LaMnO3 an antiferromagnetic insulator (AF).This phase diagram obviously shows the competition between charge, spin andorbital order in the case of manganites.

1.3 Hubbard Model 17

1.3 Hubbard Model

How should one describe such systems? A complete Hamiltonian would com-prise the kinetic energy of each nuclei, the kinetic energy of each electron, theCoulomb interaction between the nuclei, the Coulomb interaction between theelectrons and the nuclei and the Coulomb interaction between all electrons.As I want to describe macroscopic effects, in which at least 1023 particles areinvolved, the mentioned Hamiltonian is supposed to describe everything fromplasma physics at high temperatures, every possible composition of the parti-cles including solids and molecules, to low temperature collective phenomenalike superconductivity and magnetism. As it is impossible and not intended tosolve the emerging equations, the problem has to be drastically simplified: Asolid consists of ions and electrons forming a three dimensional crystal. Sincethe ions are much heavier than the electrons, the velocity of the the nucleiis of the order 10−4 − 10−5 smaller than the velocity of the electrons, whichmotivates to think of the ions as forming a rigid lattice. This ansatz is de-noted as Born-Oppenheimer approximation [37]. With this approximation onecompletely rules out the possibility of melting and other lattice effects. Thus,structural changes as can be seen in manganites will not be taken into accountin this work. Also phonons arising from lattice vibrations [37] are neglected inthis thesis.A model suitable for describing strongly interacting electrons on a lattice is theHubbard model [38–40]. One way of motivating it, starts in the single electronpicture trying to identify low temperature degrees of freedom and to extractinteraction parameter ab initio. Therefore one attempts to solve for the one-electron band-structure for the given periodic potential formed by the crystalstructure. A method able to do so is the Density Functional Theory [41–43].From the bands near the Fermi energy, which are the d-bands in transitionmetal oxides being most important for the low temperature physics, one cannow construct Wannier functions. These states are well localized around eachnucleus. With these states one can calculate the proper values for the electron-electron interaction [44–49] and set up a model Hamiltonian for the d-bands.I will now present another way for setting up a model Hamiltonian [50,51] fordegenerate bands. This way is a bit less rigorous, but it helps understandingthe underlying principles. The complete Hamiltonian is the sum of the singleparticle Hamiltonian HT and the interaction part HU

H = HT +HU .

One starts from the d-orbital wave functions of the transition metal Ψi,m,σ(r),where i labels the site, m labels the orbital quantum number, σ the spin quan-tum number, and r corresponds to the position. The corresponding secondquantized operator c†i,m,σ creates an electron with this wave function. One can


now write the single particle Hamiltonian as

HT =∑

ij

∑

m

∑

σ

tij,mc†i,m,σcj,m,σ,

tij,m =

∫

d3rΨ∗i,m,σ(r)

(

− ~2

2m∇2 + V (r)

)

Ψj,m,σ(r). (1.1)

A problem with this approach is that it is difficult to determine the effectivepotential V (r). If one assumes it to be the periodic potential of the nuclei, tijwill be negligible, because the transition metal atoms are far apart from eachother. The exchange between different sites i and j is mediated by the oxygenp-orbitals. Thus, if one really wants to calculate the coupling between differentsites, one has to include hopping processes mediated by the oxygen into thepotential V (r). The usual and simplest way is to guess some value for tij .Typical values are less than 1 eV taking into account experiments on transitionmetal oxides.The electron-electron interaction can be written as

HU =∑

i

∑

m1...m4

∑

σ1...σ4

Um1...m4c†i,m1,σ1

c†i,m2,σ2ci,m3,σ3

ci,m4,σ4

Um1...m4=

∫

d3r1

∫

d3r2Ψ∗m1

(r1)Ψ∗m2

(r2)U(r1 − r2)Ψm3(r2)Ψm4

(r1).

(1.2)

Here as well, one cannot simply write U(r1 − r2) ∼ e2

|r1−r2| as the bare Coulombinteraction, as one would neglect the screening effects of the remaining s- and p-electrons. If one would calculate U with the bare Coulomb interaction one findsvalues of about U ≈ 20 eV [9], which are too large to describe real experiments.From experiments one can extract typical values of U = 4− 8 eV for transitionmetal oxides [9].The two particle interaction can be assumed to be purely local, because thedirect overlap between d-orbitals of different sites is negligible and screeningeffects reduce the interaction range. In equation (1.2) are several differentmatrix elements included. Due to the symmetry of the d-orbitals only evenpowers of the wave functions Ψm lead to non-vanishing matrix elements [51].The two particle interaction thus reads

HU =∑

i

∑

m

Umni,m,↑ni,m,↓

+∑

i

∑

l<m,σ

Ulmni,l,σni,m,−σ

+∑

i

∑

l<m,σ

(Ulm − Jlm)ni,l,σni,m,σ

−∑

i

∑

l<m,σ

Jlmc†i,l,σci,l,−σc

†i,m,−σci,m,σ

+1

2

∑

i

∑

l,m,σ

Jlmc†i,l,σc

†i,l,−σci,m,−σci,m,σ

1.3 Hubbard Model 19

Um =

∫

d3r1

∫

d3r2|Ψm(r1)|2U(|r1 − r2|)|Ψm(r2)|2

Ulm =

∫

d3r1

∫

d3r2|Ψl(r1)|2U(|r1 − r2|)|Ψm(r2)|2

Jlm =

∫

d3r1

∫

d3r2Ψ∗l (r1)Ψ

∗m(r2)U(|r1 − r2|)Ψl(r2)Ψm(r1).

The operator n = c†c is the density operator for an electron. Since the interac-tion is local and spin independent, the position index i and the spin index σ areneglected in the integrals. If one now assumes that the interaction parametersare the same for all d-orbitals, writing U = Um, U ′ = Ulm and J = Jlm, onecan further collect terms and ends up with

HU = U∑

i

∑

m

ni,m,↑ni,m,↓ +

(

U ′ − 1

2J

)

∑

i

∑

l<m,σ,σ′

ni,l,σni,m,σ′

−2J∑

i

∑

l<m

~Si,l · ~Si,m + J∑

i

∑

l,m

c†i,l,↑c†i,l,↓ci,m,↓ci,m,↑. (1.3)

The first two terms in equation (1.3) are intra-orbital and inter-orbital density-density interactions. The third term is a spin-spin interaction between differentorbitals. It can be written as

~Sl · ~Sm =1

2(S+

l S−m + S−

l S+m) + Sz

l Szm

with spin-raising and -lowering operators S+ and S−. The last term in equa-tion (1.3) is a pair hopping term. In this work I will neglect this pair hoppingterm. This must be regarded as approximation to the real multi-orbital Hub-bard model [52] and is done for numerical reasons. When neglecting this term,one can introduce an additional conserved orbital quantum number. As I willshow later, this dramatically reduces the numerical effort. The spin-spin inter-action, third term in equation (1.3), corresponds to the ferromagnetical Hund’scoupling. Hund’s rules state that all orbitals are first single occupied with elec-trons of parallel spin. This is represented by the spin-spin interaction, whichlowers the energy if electrons occupy different orbitals with parallel spin. Ofcourse, occupying different orbitals also avoids the energetic costs of the density-density interactions. The Hund’s rule was not introduced artificially into thisHamiltonian but arises quite naturally from the general form of the two parti-cle interaction. If all five d-bands are equivalent, the following equation for theinteraction parameters holds [51]:

U = U ′ + 2J.

The multi-orbital Hubbard model [50,51] is now the sum of the non-local singleparticle term (1.1) and the local interaction term (1.3)

H = HT +HU . (1.4)

In some cases there is effectively only one d-band left at the Fermi energy. Inthis situation one can use the one-orbital Hubbard model [38–40], defined as

H =∑

ij,σ

tijc†i,σcj,σ + U

∑

i

ni,↑ni,↓. (1.5)


The interaction term now consists only of the intra-band interaction U . Madeup of only two terms the model looks very simple. But the appearance is de-ceiving. There are only very few rigorous results [53–55]. The Hubbard modelexhibits various phenomena like the metal insulator transition, antiferromag-netism, ferromagnetism and superconductivity, stating only some.The model is easy to handle in the extreme limits of vanishing hopping, t = 0, orvanishing interaction U = 0. For vanishing interaction, U = 0, one reproducesthe energy bands for the d-orbitals from which the Hubbard model can bederived. The precise band structure depends on the lattice. The physics isdetermined by non-interacting electrons which fill the bands up to the Fermienergy. For a half-filled band the system shows metallic behavior. The otherextreme case is a vanishing hopping amplitude t, but a finite and repulsiveelectron-electron interaction U > 0. There is no coupling between the atoms.The ground state is a distribution of the electrons onto the atoms, trying toavoid double occupancies, if possible, and is therefore highly degenerate. Thisground state is clearly insulating. The competition between both terms inequation (1.5), the tendency of delocalization versus localization, leads to theinteresting phenomena just mentioned above for different materials.In this thesis I will focus on the one-orbital Hubbard model (1.5) and the two-orbital Hubbard model, in which the orbital index in equation (1.3) can takeonly the values m = 1, 2. The two-orbital Hubbard model is a very goodmodel for describing the physics of a correlated eg-band. It takes into accountthe strong local electron-electron interactions, the hopping from one atom toanother, and the orbital degeneracy. More precisely, it represents the situationof transition metal oxides with octahedral symmetry, where the Fermi energylies within the eg-band. The model completely neglects the t2g-band. Thecoupling of the t2g-band to the eg-band due to Hund’s coupling can be verystrong as in the case of manganites [33]. Furthermore, the model does notinclude any coupling between the electrons and the lattice. The lattice, givenby the hopping parameter tij , is fixed and there is no coupling to phononscreated by lattice vibrations. So one cannot expect to see all the aspects oftransition metal oxides. Nevertheless, as I will show in this work, the two-orbital Hubbard model shows a very interesting ground state phase diagramwith different ordered phases. It is the basic model for investigating the physicsof strongly correlated electron systems including orbital degeneracy [50,51].

1.4 Exchange Interactions

After introducing the model and its parameters I will shortly discuss two veryimportant exchange mechanisms between different transition metal atoms. Theexchange is mediated by the oxygen p-orbitals. Oxygen has the strong tendencyto fill its p-orbitals to establish a completely filled shell like Neon [10]. Thus,one can assume that after an electron has moved from an oxygen atom to atransition metal atom, immediately another electron will fill up this p-shell ofthe oxygen. This results in a rather weak hopping amplitude t between thetransition metal d-orbitals. The resulting coupling between the two transition

1.4 Exchange Interactions 21

metal atoms is called indirect exchange. One remarkable point is, that althoughthere is no magnetic interaction between both transition metal atoms, there isa tendency for aligning the spins of the electrons either parallel or antiparalleldepending on the precise situation. As I will show now, this alignment is onlydue to the Coulomb interaction and the Pauli exclusion principle. The mostimportant processes in transition metal oxides are called super exchange anddouble exchange.

Super Exchange

I will firstly address the super exchange [56, 57]. It is called like this, becauseit extends the normally short range direct exchange to a longer range. For abetter understanding, I will look at the relevant states of two transition metalatoms. The situation is such that there is only one possible state at each ofthe two atoms. Altogether there are two electrons in the system. Thus we canassume that the basis for this problem consists of 6 states, reading

|1〉 = | ↑〉1| ↑〉2, |2〉 = | ↓〉1| ↓〉2, |3〉 = | ↑〉1| ↓〉2,|4〉 = | ↓〉1| ↑〉2, |5〉 = | ↑↓〉1|0〉2, |6〉 = |0〉1| ↑↓〉2,

where the index at each “ket” represents the index of the atom and the arrowdenotes the alignment of the electron. The case of an empty atom is denotedas |0〉. I assume that there is a strong but finite repulsive interaction withamplitude U , if two electrons are situated at the same atom. This is the case forthe states |5〉 and |6〉. Secondly, there is a possible hopping of an electron fromone atom to the other with amplitude −t. There are no spin flip interactions.Thus the Hamiltonian matrix can be written as

H =

0 0 0 0 0 00 0 0 0 0 00 0 0 0 −t −t0 0 0 0 −t −t0 0 −t −t U 00 0 −t −t 0 U

.

While the parallel configurations, states |1〉 and |2〉, have ground state en-ergy E = 0, there is one mixed state, mainly consisting of the states withspin up at one atom and spin down at the other atom, which has energyE = 1

2U − 12

√U2 + 16t2 ≈ −8 t2

U for large U . This energy gain of an antipar-allel spin state is called super exchange, and can lead to an antiferromagneticground state in an extended system, where all nearest neighbor electrons arealigned antiparallel. A well known example for a transition metal oxide withantiferromagnetic structure due to super exchange is NiO [1].

Double Exchange

Indirect exchange can also lead to a ferromagnetic coupling. A first clear signwas found in 1950 in the manganite compound La1−xCaxMnO3 [58]. The resultscould be explained by the indirect exchange called double exchange [59–61]. The


Figure 1.7: Typical situation of a double exchange process. For each transition metalatom “T” there is one localized magnetic moment and one itinerant level. Oxygen “O”acts as intermediary for the hopping.

situation is such that there is a localized moment at each transition metal atomand one electron, which can move from one atom to another by hybridizationthrough the oxygen atom, see figure 1.7. One can imagine that the localizedmoments are formed by a half-filled t2g-shell and the itinerant electron occupiesan eg-state. An important point is that there is only one itinerant electron fortwo transition metal atoms, which couples locally via Hund’s coupling to thelocal moments. As there are no spin flip processes, I assume that the singleelectron is an up-electron. A basis for this problem is given by

|1〉 = | ⇑, ↑〉1| ⇑, 0〉2, |2〉 = | ⇑, 0〉1| ⇑, ↑〉2, |3〉 = | ⇓, ↑〉1| ⇑, 0〉2,|4〉 = | ⇓, 0〉1| ⇑, ↑〉2, |5〉 = | ⇑, ↑〉1| ⇓, 0〉2, |6〉 = | ⇑, 0〉1| ⇓, ↑〉2,|7〉 = | ⇓, ↑〉1| ⇓, 0〉2, |8〉 = | ⇓, 0〉1| ⇓, ↑〉2,

where the double arrow represents the localized moment and the small arrowthe electron which is either at atom 1 or atom 2. The Hamilton matrix for thisproblem reads,

H =

−J −t 0 0 0 0 0 0−t −J 0 0 0 0 0 00 0 J −t 0 0 0 00 0 −t −J 0 0 0 00 0 0 0 −J −t 0 00 0 0 0 −t J 0 00 0 0 0 0 0 J −t0 0 0 0 0 0 −t J

.

The ground state of this system is a combination of the first two states withenergy E = −J − t, where J > 0 is the Hund’s coupling and t the hoppingamplitude. This state is clearly a “ferromagnetic” state, as both moments andthe electron point into the same direction.

1.5 Types of Magnetic Order

These very simplified models already show that depending on the exact sit-uation magnetic moments may align parallel or antiparallel. In general, for

1.6 Outline 23

Figure 1.8: Different possible magnetic orders for a simple cubic lattice. Red arrowscorrespond to up electrons and blue arrows to down electrons. See text for explanation.

real materials more complex arrangements are possible. The phase diagram ofLa1−xCaxMnO3, chapter 1.2.4, is a characteristic example. Figure 1.8 showssome common spin arrangements for a cubic lattice [29]. The F-type denotes anordinary ferromagnetic state, while in the A-type electrons in one layer are or-dered ferromagnetically but the layers itself are ordered antiferromagnetically.In the C-type the moments lying along the z-direction are aligned ferromagnet-ically. The G-type order represents a Neel-state in which all nearest neighborsare aligned antiparallel. Nevertheless, these four types are only very simpleexamples. In nature the situation if often more complex involving charge, spinand orbital degrees of freedom. In figure 1.8 only situations were shown wherethe moments are parallel or antiparallel. But the direction of the moments maychange by a smaller angle than 180 leading to a canted state or an order with aperiodicity larger than two. In the latter case one speaks of spin density waves.

1.6 Outline

The further outline of this work is as follows. In the next two chapters I willexplain the theoretical foundations of the later calculations. I will introduce thedynamical mean field theory and the Bethe lattice, for which I performed thecalculations. As the dynamical mean field theory relates the lattice problemonto a quantum impurity model, I will present in chapter 3 the numericalrenormalization group and the density matrix renormalization group, whichboth can be used for solving impurity models.After introducing their theoretical concepts in chapter 3, chapter 4 directlyaddresses a comparison of both impurity solvers for some selected situations,which must be handled when trying to calculate magnetic phases within the


dynamical mean field theory.Finally, chapter 5 and 6 are dedicated to the results for the magnetic phasesof the Hubbard model. Chapter 5 deals with the one-orbital situation. Thepossible magnetic states are analyzed over a wide range of the interaction pa-rameters also varying the form of the electron hopping tij . The form of thehopping parameter is changed by introducing a next nearest neighbor hoppingterm, which has influence on the metal insulator transition, the antiferromag-netic Neel-state, and the possible ferromagnetic states. Chapter 6 analyzes themagnetic properties of the two-orbital Hubbard model. There one can observethe competition of the just mentioned super exchange and the double exchangefor fillings of 〈n〉 ≈ 1.5. Besides this, quarter filling represents a situation, whereone can observe a clash of several different phases corresponding to the spin,charge, and orbital degrees of freedom. Finally, I will summarize this work andgive a short outlook.

CHAPTER 2

Dynamical Mean Field Theory

2.1 Introduction

This chapter gives an introduction to the Dynamical Mean Field Theory, whichI have used for solving the lattice model. I will discuss the mathematical deriva-tion of the method and the approximations made. As the main goal of this workis the calculation of magnetic phase diagrams of the Hubbard model and prop-erties of the system within these phases, I will show how to stabilize magneticordered solutions. In the last section of this chapter, I will introduce the latticefor which the calculations were performed.

2.2 Cavity Construction

For the sake of simplicity of the derivation, I will use a simple example. So letme first introduce the Ising model [62,63]. Let there be a number of independentvariables σi = −1, 1. One can assume these variables to be spins on a lattice.The Hamiltonian for this model reads

H =1

2

∑

i,j

Ji,jσiσj − h∑

i

σi.

The constants Ji,j represent a coupling between the spins, and h is a homo-geneous magnetic field acting on all spins. The property, which completelydescribes the physics and behavior of the system in equilibrium for a giventemperature T , is the free energy F or the partition function Z, respectively,

Z(β, Ji,j , h) = Trσ

exp (−βH)

F (β, Ji,j , h) = − 1

βlogZ,

26 Dynamical Mean Field Theory

where β = 1/(kBT ). Expectation values and correlation functions can be easilycalculated by derivatives of the free energy. For example the magnetization,given by M =

∑

i Tr σ(σi exp (−βH))/Z, can be written as

M = −dF (β, Ji,j , h)

dh.

The magnetization represents an order parameter for the ferromagnetic phasetransition occurring in this model [62, 63]. The Ising model at T = 0 withferromagnetic coupling Ji,j < 0 between nearest neighbors only is clearly fer-romagnetic. The lowest energy configuration is such, that all spins are alignedin the same direction. At this point the system is Z2 symmetric, meaning thatthe ground state is twofold degenerate. The value of all spins is simultaneouslyeither σ = 1 or σ = −1. At very high temperatures, Ji,j ≪ kBT , the systemwill be in the paramagnetic state, where all spins are in principle decoupledand thus M = 0. The existence of a finite temperature, T > 0, below whichthe spins begin to order, critically depends on the dimension of the system. Fordimension d = 1, the spins align only for T = 0 [62, 63], while for d = 2 thesystem exhibits a finite temperature phase transition, as found by Onsager [64].Already for the three dimensional system there is no analytical solution to thisproblem.For strongly interacting electrons on a lattice, as introduced in chapter 1.3, thesituation is even more difficult. Nevertheless, there are numerical and analyticalmethods to study strongly correlated electron systems. A numerical methodthat is able to directly simulate a lattice model is Quantum Monte Carlo [65–67]. But Quantum Monte Carlo is limited to small systems and simulationsof rather high temperatures due to the computational effort. Besides this,there are parameter regions where the simulations will fail because of the sign-problem [68, 69]. There are also other approaches like exact diagonalization,which is limited to even smaller clusters. These examples are by no means allin the zoo of possible methods for such systems. I wanted to illustrate thatthere are different methods all having advantages and disadvantages. Thereis no best method for the model I am intending to study. An overview aboutanalytical and numerical methods for interacting quantum systems focusing onmetal insulator transitions can be found in M. Imada [14].In this work I use the dynamical mean field theory. Using this approach it ispossible to scan through the whole parameter region, temperature and interac-tion parameters. I am able to identify and analyze different phases. Of course,this method has also some drawbacks. Besides the approximations, which I willstate below, there are also sometimes problems stabilizing ordered phases. Iwill come back to this point later.The aim of any mean field theory is to relate the lattice problem to a pure localproblem, called impurity problem, by partially tracing out all degrees of freedombut one site, here denoted as “site 0”. This is called a cavity construction [70].For this purpose, one splits the trace of the partition function into a partcontaining only degrees of freedom of “site 0”, Tr 0, and one part containingthe rest of the system, Tr C . The Hamiltonian is split into three parts: H0

containing only parts of the single site 0, HC0 parts which connect the single

2.2 Cavity Construction 27

site to the rest of the system and HC the rest of the system, which containsno degrees of freedom from the single site. The partition function for the Isingmodel can then be written as

Z = Tr exp(−βH)

= Tr0

[

exp(−βH0) TrC

[

exp(−βHC0 ) exp(−βHC)

]

]

,

since it denotes a classical model. Expanding exp(−βHC0 ), one finds

Z = Tr exp(−βH)

= Tr0

[

exp(βhσ0) TrC

[

exp

−β 1

2σ0

∑

i6=0

(J0i + Ji0)σi

exp

−β

1

2

∑

i,j 6=0

Jijσiσj − h∑

i6=0

σi

]]

= Tr0

[

exp(βhσ0) TrC

[ ∞∑

n=0

1

n!

−β 1

2σ0

∑

i6=0

(J0i + Ji0)σi

n

exp

−β

1

2

∑

i,j 6=0

Jijσiσj − h∑

i6=0

σi

]]

.

If one now performs the trace over the system without site 0, one can performa cumulant expansion and obtains

Z = Tr0

[

exp

(

βhσ0 +

∞∑

n=1

(σ0)n

n!

∑

i1...in

(−β)n

(

n∏

k=1

1

2(J0ik + Jik0)

)

〈σi1 . . . σin〉Ccum

)]

.

Here 〈·〉Ccum denotes the cumulant within the cavity system. Until now every-thing is exact. But nothing was gained, as the new action contains all powersof σ0 and infinite many cumulant expectation values have to be calculated.The first approximation is to assume that the expectation value in the systemwithout site 0 is the same as in the system with site 0, so 〈·〉Ccum = 〈·〉cum. Thenext step is to neglect all terms in the sum for n > 1. Both approximationscan be justified by an infinite coordination number z and an infinite extendedlattice. One ends up with the following formula

Z = Trσ0

exp

(

−βσ0

(

−h+1

2

∑

i

(J0i + Ji0)〈σi〉))

,

where 〈σi〉cum = 〈σi〉 holds. Analyzing a homogeneous system with interactionsbetween z nearest neighbors only

J0i = Ji0 =

J i is nearest neighbor of 00 else

,


the partition function for the Ising model in the mean field approximation reads

Z = Tr [exp(−βσ(−h + zJ〈σ〉 ))] ,

where 〈σ〉 has to be calculated self-consistently. It can be shown that thisformula is exact in the limit of an infinite dimensional lattice, in the sense ofan infinite coordination number z. There one has to scale zJ → J⋆ = const forz → ∞ [70]. Calculating the expectation value 〈σ〉 finally yields the familiar self-consistency equation in the Weiss molecular field theory of the ferromagnet [71]

m = tanh (−β(−h+ J⋆m)) .

A spontaneous magnetization, h = 0 but m 6= 0, can occur for β|J⋆| > 1 leadingto a finite temperature phase transition at T = |J⋆|/kB .

2.3 Dynamical Mean Field Theory

The next step is to formulate this theory for a quantum mechanical system likethe Hubbard model [70, 72–80]. I will present a derivation for the one-orbitalHubbard model, as the calculations do not change for a multi-orbital Hubbardmodel. The main difference between both Hamiltonian is the local interactionpart, which is absorbed unchanged into the effective action derived. The par-tition function must now be written as a functional integral over Grassmannvariables [81]

Z =

∫

∏

i

Dc†i,σDci,σ exp(−S)

S =

∫ β

0dτ

∑

i,σ

c†i,σ(τ)∂τ ci,σ(τ) +∑

ij,σ

tijc†i,σ(τ)cj,σ(τ)

−µ∑

i,σ

ni,σ(τ) + U∑

i

ni,↑(τ)ni,↓(τ)

.

The imaginary time argument τ of the Grassmann variables will be droppedin the following formulas. With respect to a single site 0, one again splits upthe action into the part of the single site S0, the part connecting the single sitewith the lattice SC

0 , and the rest of the action SC . Expanding the action in theSC

0 part, one again obtains an effective action for the single site [70]

Seff =

∫ β

0dτ

(

∑

σ

c†0,σ(∂τ − µ)c0,σ + Un0,↑n0,↓

)

+∑

σ

∞∑

n=1

1

n!

∑

i1...jn

∫ β

0dτi1 . . . dτjn

(

t0i1c†0,σ . . . t0inc

†0,σ

GC,coni1...jn

(τi1 . . . τjn)tj10c0,σ . . . tjn0c0,σ

)

.

2.3 Dynamical Mean Field Theory 29

Instead of the expectation values 〈σi1 . . . σin〉 in the Ising model, the effectivefield is now given by the connected Green’s functions of the cavity systemGC,con. One can now perform the same approximations as above and neglectall terms with n ≥ 2 in this action. This results in an effective field, which onlycontains the one-particle Green’s function, see textbook definition [82],

GC(τ − τ ′) = −⟨

T c(τ) c†(τ ′)⟩

C

GC(iωn) =

∫ β

0exp(iωnτ)G

C(τ)

GC(ζ) =1

ζ − Σ(ζ)(2.1)

in the action of the cavity Hamiltonian 〈·〉C . The third line in equation (2.1)follows from analytic continuation of the the second line introducing the self-energy of the system Σ(ζ) [81]. I can neglect the subscript “con”, as the usualone-particle Green’s function equals the definition of the connected one-particleGreen’s function. The effective action is now given by

Seff =

∫ β

0dτdτ ′

∑

σ

c†0,σ(τ)

(∂τ − µ)δ(τ − τ ′)+∑

ij 6=0

ti0tj0GCij(τ − τ ′)

c0,σ(τ ′)

+

∫ β

0dτUn0,↑(τ)n0,↓(τ).

The effective field GCij(τ) depends on the imaginary time τ , hence the name

dynamical mean field theory. Although this problem contains only degrees offreedom of a single site, there is no explicit solution for arbitrary GC

ij . Thiseffective single site problem corresponds to a quantum impurity problem. Therelation between this effective action and that of a quantum impurity model isshown in the next chapter. There I will also discuss two numerical approachesto obtain the one-particle Green’s function for such models.At this point I will bring this effective action into a more appropriate formintroducing the effective field G:

Seff =: −∫ β

0dτdτ ′

∑

σ

c†0,σG−1(τ − τ ′)c0,σ + Un0,↑n0↓.

Fourier transformation and analytic continuation with respect to τ of the non-interacting part of this action yields

G−1(ω + i0) := ω + i0 + µ−∑

ij

t0it0jGCij(ω + i0).

The shape of the lattice only enters in the sum over all Green’s functions∑

ij t0it0jGCij(ω+ i0). Doing the Fourier transformation in the lattice sites [70],

one ends up with

G−1(ω + i0) = Σ(ω + i0) +

(∫

dǫρ(ǫ)

ω + i0 + µ− Σ(ω + i0) − ǫ

)−1

(2.2)


where ρ(ǫ) is the density of states of the non-interacting lattice problem. Oneuses here that the self-energy is purely local Σij = Σδij [75–77]. The completelattice structure and all hopping terms are included in ρ(ǫ). This integral is theonly point in the DMFT, where the lattice structure enters.As one cannot calculate the Green’s function (2.1) explicitly for an arbitraryimpurity action, I will solve the self-consistency equation (2.1,2.2) by an itera-tive procedure. Starting with an arbitrary self-energy, which can be zero, onecalculates the effective action and solves the corresponding impurity problemfor a new self-energy. This must be repeated until the self-energy does notchange anymore. This procedure is shown in figure 2.1.Summarized, one has done the following approximations: One neglects all termsof higher order than one in the expansion of the effective action, and one assumesthe self-energy to be local. One should ask, how good are these approximations?Are they controlled and is there a limit, where these equations are exact? Theanswer to the last question is affirmative. In the limit of infinite dimensions orinfinite coordination number, the dynamical mean field theory becomes exact.The crucial point in this limit is that one has to scale the hopping parametertij as

tij√z|i−j| −→ t⋆ij

to get a non-trivial solution for the physical properties of the system [75]. Takingthe limit z −→ ∞, one can proof that all non-local contributions to the self-energy vanish and that the truncation of the terms in the effective action isexact [75].The DMFT must be considered as an approximation for real materials. Never-theless, calculations showed that within the DMFT a large amount of physicalproperties of, for example, transition metal compounds can be analyzed, andalso compared to experiments at least qualitatively, sometimes even quantita-tively [83–87]. The actual structure of the lattice does appear in the DMFTequation only through the local density of states. It has a profound effect onthe details of the resulting physics. In particular, different lattice types arestill discriminated by the actual form of the DOS and asymmetries give riseto competitions between interactions. Therefore the DMFT is a good startingpoint to analyze and understand the very difficult many body effects seen inreal materials.

2.4 Magnetic solutions

2.4.1 Ferromagnetism

In this work I will concentrate on magnetically ordered states. With the DMFTI am able to solve the lattice problem and obtain the lattice Green’s functionor self-energy. In the above equations the effective field G (2.2) was completelyspin independent. Due to the SU2 symmetry of the Hamiltonian, one cannotexpect to obtain a magnetically ordered state as solution. To achieve this,one has to calculate independent effective fields for the spin-up and spin-down

2.4 Magnetic solutions 31

Impurity

Bath

Cavity-ConstructionProjecting the

impurity self-energy

onto the lattice

Figure 2.1: Iterative DMFT-loop. The lattice is projected onto an impurity problem.The solution of the impurity calculation is projected back onto the lattice. This isrepeated until convergence.


components [70]

G−1σ (iωn) = Σσ(iωn) +

(∫

dǫρ(ǫ)

iωn + µ− Σσ(iωn) − ǫ

)−1

.

Additionally, it is important to break the SU2 symmetry of the system in thevery first iteration by applying a small magnetic field [70]. After the firstiteration I switch off the magnetic field and continue as described in figure 2.1.If the system possesses a ferromagnetic instability, it will evolve into a self-

consistent solution with Σ↑ 6= Σ↓ and a polarization p =|n↑−n↓|n↑+n↓

> 0. The

occupations can be calculated from the self-energy by

nσ = − 1

π

∫

dωf(ω)Im

∫

dǫρ(ǫ)

ω + i0 + µ− Σσ(ω) − ǫ,

where f(ω) denotes the Fermi-function.

2.4.2 Antiferromagnetism

I will also present solutions with antiferromagnetic long-range order. The im-portant point is, that this ordered state is non-homogeneous; neighboring sites,A and B, have different self-energies [70, 73]. Focusing on the Neel-state, thefollowing relation holds

ΣA,σ = ΣB,−σ. (2.3)

To derive the proper equation for the effective field, one has to assume that thelattice is bipartite. This means that one can divide the lattice into two sublat-tices, and each site of the A-sublattice has only B-sites as nearest neighbors.Then one can show that the self-consistency condition still holds, but now inmatrix form of the two sublattices [70,73]. The Green’s function entering mustbe calculated by matrix inversion

G(ζA, ζB) =

∫

dǫρ(ǫ)

((

ζAσ 00 ζBσ

)

− HT

)−1

=

∫

dǫρ(ǫ)

(

ζAσ −ǫ−ǫ ζBσ

)−1

. (2.4)

The matrix HT represents the hopping term between the sublattices. Usingequation (2.4) one can now calculate the new effective fields for both spin com-ponents and sublattices. As relation (2.3) holds for the two sublattices, one stillneeds to solve only one impurity problem per DMFT iteration, as interchangingspin labels yield the solution for the other sublattice.

2.5 Bethe lattice

In this last section of the chapter, I will introduce the Bethe lattice, for whichI have performed the calculations. The lattice itself is very artificial. Theconstruction of the Bethe lattice with coordination number z is very easilydone in shells:

2.5 Bethe lattice 33

• Begin with drawing one site, which is shell 0.

• Connect the first site with z neighboring sites, building shell 1.

• Connect all sites of the last shell with z − 1 new sites, building the newshell.

• There are no closed loops.

Because of this symmetric construction of the shells and the important pointthat there are no closed loops, implying that there is only one way from site A tosite B, many models can be solved exactly for the Bethe lattice [88]. However,for the Hubbard model this is in general not the case, but one can solve themodel without two particle interaction U = 0 [70]. Moreover, one is able tocalculate the density of states (DOS) of the lattice, which is essential for theDMFT. As the Bethe lattice is bipartite, I am able to perform antiferromagneticcalculations.The DOS ρ(ǫ) is given by the imaginary part of the local Green’s function asρ(ǫ) = −1/πImG(ǫ+i0). The hopping parameter is assumed to be independentof the site indices and non-vanishing only between nearest neighbors, tij = t.The local Green’s function can be written as [89]

G−100 (ζ) = ζ − t2

∑

i∈NN

G(0)ii (ζ).

G(0) is the Green’s function for the lattice, where site 0 is removed. But for

infinite coordination number G(0)ii (ζ) = Gii(ζ) = G00(ζ) holds and is indepen-

dent of the lattice site. Performing the appropriate scaling t =: t⋆√z

yields

G−100 (ζ) = ζ − (t⋆)2G00(ζ). One can now simply solve for G := G00, obtaining

G(ζ) =1

2(t⋆)2

(

ζ +√

ζ2 − 4(t⋆)2)

ρ(ω) =1

2π(t⋆)2

√

4(t⋆)2 − ω2. (2.5)

Thus the DOS of the Bethe lattice (2.5) with nearest neighbor hopping only isa semi-ellipse. One should note, that the Hilbert transformation from equation(2.2) can be performed by inserting the argument into the local Green’s function(2.5).

2.5.1 Bethe lattice with next nearest neighbor hopping

DMFT calculations depend sensitively on the DOS. The shape of the DOS canbe tuned by introducing next nearest neighbor hopping, as shown in figure 2.3.I will now derive the DOS for the Bethe lattice with nearest neighbor (NN) andnext nearest neighbor (NNN) hopping. I will call the hopping operator betweenNN-sites H1 and the hopping operator between NNN-sites H2. One can expressthe NNN-hopping term through the NN-hopping term. In infinite dimensionsone obtains [90,91]

H2 = (H1)2 − 1.


Figure 2.2: Upper panel: Part of the Bethe lattice with coordination number z = 4.The colors reveal the bipartite structure of the lattice. Lower panel: Bethe latticeincluding next nearest neighbor bonds. In the lower panel it becomes apparent thatthe next nearest neighbor bonds act between sites with the same color (sites of thesame sublattice). This introduces frustration to the Neel-state.


This equation states that the eigenstates of the Hamiltonian H = H1 + H2

equal that of H1 alone. Let t1 be the amplitude for the NN-hopping and t2 forthe NNN-hopping. The local Green’s function can thus be evaluated to [90,91]

Gt1,t2(ζ) =

⟨

0

∣

∣

∣

∣

1

ζ −H

∣

∣

∣

∣

0

⟩

=

∫

dǫρ(ǫ)

ζ − t1ǫ− t2(ǫ2 − 1)

=1

2t2b(ζ)[G(a + b(ζ)) −G(a− b(ζ))]

ρ(ǫ) =1

2π

√

4 − ǫ2

a =−t12t2

b(ζ) =√

ζ/t2 + a2 + 1. (2.6)

The third line in equation (2.6) was gained by partial fraction decompositionof the second line. G(ζ) (without subscript) represents the Green’s functionfor t2 = 0 given by equation (2.5). The DOS of the system without and with

-1 -0.5 0 0.5 1ω/W

0

1

2

3

ρ(ω

)W

t2/t

1=0

t2/t

1=0.1

t2/t

1=0.2

-0.5 0 0.5 1ω/W

t2/t

1=0.3

t2/t

1=0.6

t2/t

1=1

Figure 2.3: DOS in infinite dimensions for the Bethe lattice with NN-hopping t1 andNNN-hopping t2. The left panel shows the situation for t2

t1< 1

4, in which no singularity

is present. The right panel shows t2t1

≥ 1

4with a singularity at the lower band edge.

The functions are normalized with the bandwidth W .

next nearest neighbor hopping can be seen in figure 2.3. The black curve in theleft panel shows the semi-elliptic DOS of the Bethe lattice with NN-hoppingonly, while the other curves correspond to a finite t2 > 0. When examining the


-4

-2

0

2

4

-0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0

ω

t2/t1=0.2

-4

-2

0

2

4

-0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0

ω

t2/t1=0.25

-4

-2

0

2

4

-0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0

ω

t2/t1=0.3

Figure 2.4: The Green’s function for NN- and NNN-hopping is given by a sum oftwo Green’s function for NN-hopping only. These three plots show the arguments ofthe NN-Green’s function for different values of NNN-hopping. (See also the text forexplanation) The black and the red curves are (a + b(ω))2 − 4 and (a − b(ω))2 − 4,respectively. The NN-hopping is in all three panels t1 = 0.05. The blue vertical linerepresents the singularity at ω3 while the green vertical line represents the position ofthe roots of both functions at ω2.


spectral function, ρ(ω) = − 1πImGt1,t2(ω+ i0), one finds the following three key

values for ω

ω1 = 3t2 + 2t1

ω2 = 3t2 − 2t1

ω3 = − t214t2

− t2. (2.7)

Equation (2.6) has only a non-vanishing imaginary part, if (a+ b(ω))2 − 4 < 0or (a−b(ω))2−4 < 0 holds, which are the square root arguments of the Green’sfunction for NN-hopping entering. Both functions can be seen in figure 2.4. Thefrequency ω1 is the upper frequency at which the imaginary part of equation(2.6) vanishes. This means that ρ(ω) is zero for ω > ω1. The vertical lines infigure 2.4 represent ω2 and ω3. At frequency ω2 there is a root in the arguments.For t2

t1< 1

4 (upper panel in figure 2.4) both functions are positive for ω < ω2

resulting in a vanishing imaginary part of equation (2.6) and thus vanishingspectral function. Frequency ω3 is the lower end of both functions. For ω < ω3

both functions become imaginary, which does not lead to a finite contribution tothe spectral function, as it cancels out. For t2

t1> 1

4 both functions are shifted tonegative values and there is always a negative argument for ω3 ≤ ω ≤ ω2. Thusthe lower band edges of the spectral function is given by ω2 (ω3) for t2

t1< 1

4

( t2t1> 1

4). The frequency ω3 is exactly the point where b(ω3) = 0. Looking

in equation (2.6) one sees that there is a 1b(ω) -term resulting in a square-root

singularity for t2t1> 1

4 at ω = ω3. This can be seen in figure 2.3. In the left

panel the situation t2t1< 1

4 is plotted. There is no singularity. In the right panelthere is always a singularity at the lower band edge, exactly at ω = ω3. Fromthe considerations one can derive the bandwidth of the DOS to be

W =

4t1 t2/t1 < 1/42t1 + 4t2 + t21/(4t2) t2/t1 > 1/4

.

In the case of next nearest neighbor hopping antiferromagnetic calculationsare possible, too. The hopping Hamiltonian entering in equation (2.4) nowreads [90,91]

HT =

(

t2(ǫ2 − 1) t1ǫt1ǫ t2(ǫ

2 − 1)

)

.

As I will show later, the asymmetry and especially the singularity have greatinfluence on the DMFT calculations. Introducing the next nearest neighborhopping leads to frustration of the antiferromagnetic Neel-state and thus tocompletely different magnetic ground states.

CHAPTER 3

Impurity Solver - Theoretical

Background

3.1 The Anderson Impurity Model

In the last chapter I derived an effective action for the Hubbard model withinDMFT. This action consists of only one interacting site coupled to non-inter-acting electrons in a bath, which has to be solved self-consistently. In this chap-ter I will introduce the single impurity Anderson model [92], which is stronglyrelated to this effective action and discuss two strategies, how one can solvethis model numerically. In this work I have in particular used two matrixbased renormalization group techniques. One is the Numerical Renormaliza-tion Group (NRG) [93, 94], the other is the Density Matrix RenormalizationGroup (DMRG) [95,96]. I will explain, how both methods work and what theiradvantages and disadvantages are.The Anderson impurity model describes a magnetic atom embedded into aconduction band. It can be written as

HAnd =∑

~k,σ

ǫ~kc†~k,σc~k,σ

+∑

σ

ǫdc†d,σcd,σ + Uc†d,↑cd,↑c

†d,↓cd,↓

+∑

~k,σ

V~k

(

c†~k,σcd,σ + c†d,σc~k,σ

)

, (3.1)

where c†~k,σ(c~k,σ

) creates (annihilates) an electron in the conduction band with

quantum number (~k, σ) and energy ǫ~k. In this Hamiltonian ~k represents athree-dimensional momentum vector. The impurity itself is represented by theoperators c†d,σ (cd,σ), where ǫd is the energy level and U the amplitude for a

40 Impurity Solver - Theoretical Background

density-density interaction on the impurity. Finally, there is a hybridization be-tween the impurity and the conduction band given by V~k

. The Anderson model,and related with it the Kondo model [97–99], became famous for describing thephysics of the Kondo problem. This effect occurs, when the resistivity of, forexample, gold with embedded cobalt atoms is measured [100]. For low temper-atures an unusual increase of the resistivity is found. This physical situationcan be correctly described and understood with Hamiltonian (3.1) [93,98].The Anderson Hamiltonian is most often simplified by expanding the conduc-tion operators into spherical harmonics and assuming that only the s-wavestates are coupled to the impurity level [93,94,101,102]. Thus the Hamiltoniancan be written in the following form

HAnd =∑

σ

∫ D

−Ddǫ ǫc†ǫ,σcǫ,σ +

∑

σ


†d,↓cd,↓

+∑

σ

∫ D

−DdǫV (ǫ)

√

ρ(ǫ)(c†ǫ,σcd,σ + c†d,σcǫ,σ). (3.2)

The s-wave band of the conduction electrons is here assumed to have energiesbetween −D and D. The Hamiltonian (3.2) can be visualized as in figure 3.1.

ImpurityU , ǫd V (ǫ)

Bat

h

Ener

gyǫ

−D

D

Figure 3.1: Visualization of the Anderson model. An impurity site, with parametersU and ǫd, is coupled via a hybridization to a bath of non-interacting electrons.

In equation (3.2) I can formally integrate out the non-interacting conductionband, deriving an effective action for the impurity level [103]

Seff,And =∑

iωn

∑

σ

c†d,σ

(

iωn + ǫd +

∫

dǫV (ǫ)2ρ(ǫ)

iωn − ǫ

)

cd,σ + Uc†d,↑cd,↑c†d,↓cd,↓,

(3.3)where all operators turned into Grassmann variables. This action (3.3) has thesame form as the effective DMFT action, see chapter 2.3. I should mentionhere that ρ(ǫ) is different to the density of states of the lattice in the DMFT. Itis the DOS of the bath in equation (3.2), generated by the dispersion ǫ~k. I willnow define the coupling between the impurity level and the conduction bandas h(ǫ) := V (ǫ)

√

ρ(ǫ). Comparing with the effective action in the DMFT, one

3.2 Discretization of the Band States 41

can identify

G−10 (iωn) = iωn + ǫd +

∫

dǫh(ǫ)2

iωn − ǫ.

As G0 is an analytic function in the upper and lower complex plane, one cansolve for the coupling

h(ǫ) =

√

−ImG−10 (ǫ+ i0)

π. (3.4)

With equation (3.4), I related the DMFT self-consistency loop with the An-derson model. The Anderson model is a fully quantum mechanical many bodyproblem and there is no exact solution for calculating dynamical properties.Nevertheless, for getting an approximate solution for the Anderson model abundle of techniques are available [98].I will introduce matrix based renormalization group techniques in the next sec-tion. They are completely in the spirit of the renormalization group in statisti-cal mechanics for critical behavior. For example, in the block spin renormaliza-tion [93] one replaces small clusters of spins with only one new spin representingthe whole small cluster. Doing this iteratively one can look for fixed points inthis calculation, which describe the physics in the thermodynamic limit.A similar principle is used in the NRG/DMRG. Here, one starts with only avery small part of the whole problem, which can be diagonalized exactly. Then,by truncating the already diagonalized part and adding new degrees of freedomand repeated diagonalization, one can calculate iteratively an approximationfor the complete system.

3.2 Discretization of the Band States

Before one can apply NRG/DMRG as impurity solver one has to rewrite theHamiltonian (3.2) as an one-dimensional chain of non-interacting sites coupledto one interacting site (see figure 3.2). The Anderson Hamiltonian can be

Impurityt0 t1 t2 t3 t4

HImp HC

Figure 3.2: Single impurity Anderson model in form of a chain. The impurity site isunchanged. The non-interacting bath is rewritten to a semi-infinite chain with nearestneighbor hopping. HImp and HC correspond to equation (3.5).

written as

HAnd = HImp +Hc

HImp =∑

σ


†d,↓cd,↓

HC =∑

σ

∫ 1

−1dǫ ǫc†ǫ,σcǫ,σ +

∑

σ

∫ 1

−1dǫ h(ǫ)

(

c†ǫ,σcd,σ + c†d,σcǫ,σ

)

, (3.5)


where the conduction band energies have been scaled so that D = 1. I willnow rewrite the integral in HC as a discrete sum. For this purpose the interval[−1, 1] has been divided into disjoint intervals:

[−1, 1] =⋃

i

[ωl,i, ωu,i].

On each interval an orthonormal basis Ψi,k : [ωl,i, ωu,i] −→ R is set up so that∫

dǫΨi,k(ǫ)Ψi′,k′(ǫ) = δi,i′δk,k′. One can expand a function within this basis set.For Ψi,k, one can use normalized Legendre polynomials or a local Fourier basis.With these functions, one defines new operators

ci,k,σ =

∫

dǫ cǫ,σΨi,k(ǫ)

cǫ,σ =∑

i,k

ci,k,σΨi,k(ǫ). (3.6)

These operators follow the usual anti-commutator relations, as the basis isorthonormal [93,101,102]. One can now rewrite (3.5)

Hc =∑

k,k′,i,σ

ǫi,k,k′c†i,k,σci,k′,σ +∑

k,i,σ

hi,k

(

c†i,k,σcd,σ + c†d,σci,k,σ

)

ǫi,k,k′ =

∫

dǫǫΨi,k(ǫ)Ψi,k′(ǫ)

hi,k =

∫

dǫh(ǫ)Ψi,k(ǫ). (3.7)

Analyzing equation (3.7), one can see, that each operator ci,k,σ couples to theimpurity cd,σ . This kind of Hamiltonian is usually referred to as star geometry.For real calculations, one has to truncate the orthonormal basis set to onlya few basis functions. This must be done for assuring that the number ofsites is finite. Finally, one can tridiagonalize HC (3.7) with a Householderalgorithm [104] bringing it into chain geometry, as can be seen in figure 3.2.

3.3 Numerical Renormalization Group

3.3.1 Discretization within the NRG

The Numerical Renormalization Group (NRG) was developed by Kenneth Wil-son [93] in order to explain the Kondo effect. Some years later the methodwas applied to the Single Impurity Anderson Hamiltonian by H. Krishna-Murthy [101,102]. An overview of this method and recent developments can befound in R. Bulla et al. [94].The main point in the NRG algorithm is that one has to use a logarithmic meshfor the discretization of the conduction band. The reason for this will becomeclear below. Hence, the intervals have the form

In,± = [±Λ−(n+1),±Λ−n] n ∈ N

Ln = |Λ−(n+1) − Λ−n|.

3.3 Numerical Renormalization Group 43

The constant Λ > 1 is a free parameter, setting the logarithmic discretizationof the intervals. The goal of the logarithmic discretization is to have a verygood resolution, meaning very small intervals, near the Fermi energy ǫ = 0.Usually, one uses values for the discretization of approximately Λ ≈ 2. Asdescribed above, one sets up an orthonormal basis within each interval. ForNRG calculations one neglects all terms but the constant function in each in-terval. It can be shown that the coupling between the different orthonormalfunctions is small [93]. Nevertheless, it is a rather rude approximation, es-pecially when using Λ = 2, which is only justified a posteriori by the resultsgained [93, 94, 101, 102]. On the other hand, using only the constant functionon each interval, it can be shown that the solution is exact for Λ → 1. So it isrecommendable always to check the results by doing a calculation with smallerΛ. If nothing changes, one can assume to have a good approximation.One can now transform the conduction band Hamiltonian Hc into the formof a linear chain with the unchanged impurity at the beginning of the chain(see figure 3.2). The important point of the logarithmic discretization andthis mapping is that the coupling constants ti decrease exponentially alongthe chain. This exponential decrease is of great importance, as the iterativediagonalization is doomed to fail, if this property is not given. I will explainthis in more detail in the next section.In real calculations the chain is cut after N -sites. For NRG-calculations typicalchain lengths are of the order of N = 15 − 100 sites. Although N = 15 sitessounds very few, as the degrees of freedom grow exponentially, there are already

415 = 1073 741 824

degrees of freedom. This number is too large for a direct diagonalization, soone is forced to use iterative diagonalization schemes.

3.3.2 Iterative diagonalization

The iterative diagonalization procedure of this semi-infinite chain is visualized infigure 3.3. One begins with setting up the complete Fock space for the impuritysite. As the number of states is small, it can be diagonalized completely. Takingthe new eigenstates one can set up the basis for the impurity coupled to the firstsite of the chain. The new Fock space is just a tensor product of the eigenstatesof the impurity and a Fock space basis for the first site. Diagonalizing thissmall cluster again yields the eigenstates, which are used to set up the new basisincluding the next site. This is done iteratively as can be seen in figure 3.3. Theproblem is that the number of states in the Fock space grows exponentially. Fora complete diagonalization one is usually limited to about three or four sites ofthe lattice. In order to continue one has to truncate the Fock space. In NRGcalculation one takes the NK energetic lowest Fock space states. Only theseare used to set up the new Fock space including the next site. As a result, onecan continue to iteratively diagonalize the chain always truncating the space tothe energetic lowest NK states. The reason why this truncation scheme workshere is that the coupling of the next site to the already diagonalized part of thechain is weak compared to the current energetic scale. The coupling between


Iteration

t0 t1 t2 t3 t4

Impurity

Truncation Line

Ener

gy

Figure 3.3: Iterative diagonalization procedure in the NRG. See the text for description.

the states should be small compared to the difference between the energeticlowest and highest state. Loosely speaking, the next site is supposed to beonly a small perturbation for the already diagonalized chain. This is ensuredby the exponential decrease of the hopping amplitude ti, which comes from thelogarithmic discretization of the bath states in the beginning. This iterationprocedure can be written as

HN = Λ(N−1)/2

(

HImp + t0∑

σ

(c†d,σc0,σ + c†0,σcd,σ)

+

N∑

i=0,σ

ǫic†i,σci,σ +

N−1∑

i=0,σ

ti+1(c†i,σci+1,σ + c†i+1,σci,σ)

HN+1 =√

ΛHN + ΛN/2ǫN+1

∑

σ

c†N+1,σcN+1,σ

+ΛN/2tN+1

∑

σ

(c†N,σcN+1,σ + c†N+1,σcN,σ)

H = limN→∞

Λ−(N−1)/2HN . (3.8)

This defines a renormalization group transformation R [93, 101,102]

HN+1 = R[HN ], (3.9)

which transforms a Hamiltonian HN into another Hamiltonian HN+1. Theprefactor Λ(N−1)/2 has been introduced in equation (3.8) to compensate thedecreasing of the hopping parameters ti. Thus, the energy levels remain of thesame order of magnitude ensuring a constant numerical diagonalization error.Even if one is able to continue iteratively diagonalizing the semi-infinite chain,one has to stop at some point. There are two criteria when to stop the calcula-tion. The first criterion is given when the calculation has reached a fixed point,


meaning, that the energy levels of HN equal HN+2 and do not change anymore.The energy levels may change from even to odd iterations even when the fixedpoint is reached. When the zero temperature fixed point is reached, one canstop the iterative procedure and one is able to calculate zero temperature prop-erties. Care must be taken, as also meta-stable fixed points may exist. Theother criterion for stopping the procedure applies if one is interested in finitetemperature results. As the coupling decreases exponentially, also the imposedenergy scale decreases. At some point the changing of the energy levels will besmaller than the energy scale defined by the temperature. The changes in theposition of the energy levels by further diagonalizing the chain will not changethe Boltzmann weights in the calculation of thermodynamical properties forthe given temperature anymore. This is the point when one can stop the di-agonalization of the chain. Further diagonalization will not change the finitetemperature results.

3.3.3 Calculation of impurity properties

The calculation of static expectation values on the impurity is straight for-ward [94]. One simply sets up the matrix for the corresponding operator atthe beginning of the iterative diagonalization. This matrix is always updatedduring the procedure into the current basis. Using the energy values and thecorresponding Boltzmann weights one is able to calculate thermodynamic staticexpectation values. Let A be the operator of interest, β = 1

kBT the inverse tem-perature, Z the partition function, and |i〉 a state from the current basis withenergy Ei. Then the expectation value reads

〈A〉 =1

Z

∑

i

exp(−βEi)〈i|A|i〉.

In this formula the sum is supposed to run over a complete basis set, which isdifficult to determine, as the basis is truncated during the iterative diagonaliza-tion. For thermodynamic properties it is often a very good approximation toassume that the only relevant contributions for a given temperature come fromthe chain with the length, where the temperature equals approximately thecharacteristic energy scale of the last site [94]. A more sophisticated approachwill be shown below.The calculation of dynamical properties is a little bit more subtle. At theend of the calculation one is able to calculate static expectation values andhas found the eigenstates and the corresponding energies. Thus one is able touse Lehmann’s formula to calculate spectral functions for an operator A. Theretarded single particle Green’s function for fermionic operators is given by

GAB(t) = −iΘ(t)Tr ρ [A(t)B(0) +B(0)A(t)] . (3.10)

If there were no truncation during the diagonalization, using Laplace transfor-mation, one would obtain

GAB(ζ) =1

Z

∑

i,j

〈i|A|j〉〈j|B|i〉exp(−βEi) + exp(−βEj)

ζ + Ei − Ej,


where ρ = exp(−βH)/Z was inserted. Furthermore, the partition function isgiven by Z =

∑

i exp(−βEi), and ζ is a complex variable. The spectral functionobtained by this reads

ρAB(ω) =1

Z

∑

i,j

〈i|A|j〉〈j|B|i〉(exp(−βEi) + exp(−βEj))δ(ω + Ei − Ej).

The problematic point in this formula is that the sum over i, j must be over acomplete basis set. If one uses this formula only with the basis set of the lastiteration, one will use only a very tiny fraction of a complete basis due to thetruncation during the iterative diagonalization. Consequently, the calculatedspectral function will contain only frequency points very close to ω = 0. Allhigh energy features will be lost. But adding all iterations of the chain will givean over-complete basis. It will include a double counting of some states, whichmust be accounted for by some regulation process [105, 106]. This regulationprocedure is in some way arbitrary and introduces a new and uncontrolledapproximation.To overcome this regulation one has to identify a complete basis set. The idea,how to do this in the NRG chain, was by F. Anders and A. Schiller [107, 108].It was tested for equilibrium Green’s functions by RP, T. Pruschke and F.Anders [109] and separately by A. Weichselbaum and J. von Delft [110]. Thenew approach does focus on a Wilson chain of length N . In the beginning of theiteration procedure a complete basis can be simply identified as a tensor productof all single site Fock space states. Before the first truncation in the iterativediagonalization procedure at site m < N , the eigenstates of the diagonalizationΨm

kmyield a complete basis set of the chain of site m. If one performs a tensor

product with a basis for the rest of the chain m < k ≤ N ,

Ψmkm

⊗ ψm+1km+1

⊗ . . .⊗ ψNkN,

one still obtains a complete basis set. Here Ψmkm

is an eigenstate of the chain of

length m and ψm+1km+1

a basis state of the single site m+ 1. If one now truncatesthe chain at length m, one neglects these states times a complete basis for therest of the chain, when trying to build up a complete basis set. The part of thebasis, which is created by the kept states, the states which are not truncated,is not changed by the diagonalization. The kept states will be changed in thediagonalization of the chain of length m+ 1. But a tensor product of all statesin m + 1 with a complete basis for the Fock space for m + 1 < k ≤ N spansthe same space as the kept states at length m times a basis for the Fock spacem < k ≤ N :

Span

∑

km∈kept

∑

km+1,...,kN

Ψmkm

⊗ ψm+1km+1

⊗ . . .⊗ ψNkN

=

Span

∑

km+1

∑

km+2,...,kN

Ψm+1km+1

⊗ ψm+2km+2

⊗ . . . ⊗ ψNkN

.


Thus, one can identify a complete basis set which consists of all truncated statestimes a basis for the rest of the chain, where they have been truncated. In thelast step of the iterative diagonalization all states are supposed to be truncated.In the following, the notation for a state shall be:

|Ψ〉 = |m,km, l〉 = |m,km〉 ⊗ |l〉,

where m labels the chain site, km is the index of the eigenstates at this siteand l is an index for a state in the basis for the rest of the chain. After thetruncation of a state, denoted as |·〉Trunc, the impurity properties inherent inthis state are not changed anymore as the state is decoupled from the rest ofthe chain

Trunc〈m,km1, l1|AImp|m,km2

, l2〉Trunc = 〈m,km1|AImp|m,km2

〉δl1,l2.

Thus the environment states |l〉 do not change the expectation values of animpurity operator after the truncation. If one now inserts a complete basis setconsisting only of truncated states in equation (3.10), one question must stillbe clarified: What is the expectation value of two states truncated at differentsites m and m′ in the chain for m < m′. A state truncated at site m′ can bewritten as a linear combination of kept states at site m

|m′, k′m, l〉Trunc =∑

km∈kept

∑

l′

Φkm,l′ |m,km, l′〉,

where the l′ is labeling environment states. Φkm,l′ are coefficients in the orthog-onal matrices of the diagonalization. Thus the contributions of combinationsof states truncated at different shells m < m′ can be accounted for. One hasto take the combinations of the truncated state km in shell m and all the keptstates in the same shell, as the environment states do not change the expectationvalue with an impurity operator.Although the environment states do not change these direct expectation valueswith an impurity operator, they influence the value of the density operator.Their influence on it can be taken into account with the reduced density matrix[111]

ρredkm,k′

m=∑

l

〈m,km, l|ρ|m,k′m, l〉.

The contributions of a state to the static or dynamical expectation value of theimpurity can be calculated correctly at the site, where the state is truncated.For calculating magnetic solutions within the NRG, it is essential to use thereduced density matrix ρred [111]. A small magnetic field can strongly influencethe ground state properties of an impurity calculation. Calculating dynamicalproperties without using the reduced density matrix did often dramaticallyunderestimate the influence of the magnetic field.In summary, when doing a calculation using a complete basis set, one must takeall the contributions of all truncated states and all states of the last iteration.In the end this can be easily understood. All kept states are only refined in thefollowing iterations. Thus contribution between two kept states can be taken


into account in later iterations. Defining |m, i〉 as a state in iteration m, theGreen’s function can be written as

GAB(ζ) =N∑

m=0

∑

i,j,k

′(

〈m, i|A|m, j〉ρredj,k 〈m,k|B|m, i〉

+〈m, i|A|m,k〉ρredj,i 〈m,k|B|m, j〉

) 1

ζ + Ei − Ek

ρAB(ω) =N∑

m=0

∑

i,j,k

′(

〈m, i|A|m, j〉ρredj,k 〈m,k|B|m, i〉

+〈m, i|A|m,k〉ρredj,i 〈m,k|B|m, j〉

)

δ(ω − Ei + Ek), (3.11)

where now the summations∑

i,j,k′ is such, that at least one of the states |m, i〉

or |m,k〉 is a truncated state. The spectral function is again given by ρAB(ω) =− 1

πImGAB(ω + i0).This formula can now be applied to the NRG diagonalization. After diago-nalization of the complete chain, one sets up the density matrix and iteratesbackwards from the end of the chain, setting up the reduced density matrixfor each site. For each iteration one applies equation (3.11) and collects all thedelta peaks for the spectral function. To gain a smooth spectrum, one has toreplace the delta peaks in equation (3.11) by continuous functions. As the dis-cretization was done on a logarithmic scale, it is quite common to use Gaussianfunctions on a logarithmic scale for doing this broadening [94]

δ(ω − E) →exp

(

− b2

4

)

bE√π

exp(

− (log(|ω|/E)/b)2)

,

where b is a broadening parameter, whose value is typically b ≈ 0.3 − 0.8. Forfrequencies below the smallest energetic discretization Λ−(N−1)/2 it is advisableto use Lorentzians instead of logarithmic Gaussians. The reason is that purelogarithmic Gaussians will always result in ρ(ω = 0) = 0. As one can thinkof the smallest energetic discretization as a finite temperature, it is justified touse a Lorentzian here.The same complete Fock space treatment can be used for calculating static ex-pectation values, too [110]. One can determine the exact Boltzmann weight toeach Fock space state in the complete basis set. This will lead to the situationthat more than only one NRG shell contributes to thermodynamical proper-ties. The advantage of this procedure is, that one can calculate properties atarbitrary temperature with precise Boltzmann weights.

3.3.4 Calculating the Self-energy

For the DMFT self-consistency equation one needs the self-energy of the im-purity. The expression for the self-energy can be derived by equations of mo-


tion [112]

1 = (ζ − ǫd)Gcd,σ

,c†d,σ

(ζ) − UGcd,σ

c†d,−σ

cd,−σ

, c†d,σ

(ζ)

−∑

~k

V~kGc~k,σ

,c†d,σ

(ζ)

0 = (ζ − ǫk)Gc~k ,σ,c†

d,σ

(ζ) − V~kG

cd,σ

,c†d,σ

(ζ)

1 = (ζ − ǫd)Gcd,σ

,c†d,σ

(ζ) − UFσ(ζ) − ∆Gcd,σ

,c†d,σ

(ζ)

∆(ζ) =∑

~k

V 2~k

1

ζ − ǫ~k

Fσ(ζ) = Gcd,σ

c†d,−σ

cd,−σ

, c†d,σ

(ζ),

finally yielding

⇒ Σσ(ζ) = UFσ(ζ)

Gcd,σ

,c†d,σ

(ζ)(3.12)

with the notation of equation (3.1) and the Green’s function GA,B for operatorsA and B. Thus, the self-energy of the impurity can be calculated as a quotientof two Green’s functions, which in turn can be calculated within the schemejust derived in this chapter.

3.3.5 Two-orbital Anderson model

In the last sections I have dealt with the single impurity Anderson model.But focusing on a two-orbital Hubbard model within DMFT, it is essentialto introduce the two-orbital Anderson model, as DMFT will map the multi-orbital lattice model onto a multi-orbital Anderson model. The local part ofthe Anderson Hamiltonian is the same as in the Hubbard model. Thus theHamiltonian for the two-orbital Anderson model reads

HAnd,2 = HC +HImp

HImp =2∑

m=1

∑

σ

ǫdnd,m,σ + U2∑

m=1

nd,m,↑nd,m,↓

+

(

U ′ − 1

2J

)

∑

σ,σ′

nd,1,σnd,2,σ′ − 2J ~Sd,1 · ~Sd,2

HC =

2∑

m=1

∑

σ

∫ D

−Ddǫ ǫc†ǫ,m,σcǫ,m,σ

+

2∑

m=1

∑

σ

∫ D

−Ddǫ V (ǫ)

√

ρ(ǫ)(c†ǫ,m,σcd,m,σ + c†d,m,σcǫ,m,σ),

where each operator has an additional orbital quantum number m. Operatorsacting on the impurity are labeled with “d”. This model can be mapped, asshown above, on an one-dimensional chain, where each site has a local Fock


space consisting of 16 states due to the two degenerate conduction bands. Thusthe complete Fock space grows with the number of sites N as 16N . To getreasonable results for static and dynamic properties within the NRG, I keptup to 5000 states per iteration. For one complete diagonalization and thecalculation of spectral functions one needs up to 16GB main memory and severalprocessor hours computing time on a multi-core-processor. In contrast, for anone-orbital model often 1000− 2000 states and 1GB main memory are enough.For the calculations in the two-orbital model I used three conserved quantumnumbers: particle number N , total SZ of the spin components and total orbitalquantum number Tz. If one would include the above mentioned pair hoppingterm into the Hamiltonian the orbital quantum number would not be conservedanymore. This would result in larger quantum number subspaces during theiterative diagonalization and thus larger amount of main memory and time forthe computation.

3.4 Density Matrix Renormalization Group

I developed a new Density Matrix Renormalization Group (DMRG) code. Iespecially focused on solving impurity problems like the Anderson impuritymodel. The main reason for this development was to get some independentresults with a method different from the NRG for the magnetic phase dia-gram in the Hubbard model. DMRG was originally designed for treating one-dimensional systems [95, 96, 113, 114]. But as I have shown above, an impu-rity model like the Anderson model can be mapped onto a one-dimensionalchain [93]. In contrast to the NRG, no logarithmic discretization must be usedfor the mapping within the DMRG. For this method one can freely choose theintervals. This allows for more freedom and one is able to properly control theaccuracy of the discretization. DMRG can then be used for diagonalizing thischain representing the Anderson impurity model [115–118].

3.4.1 Iterative diagonalization

Similar to the NRG, also in the DMRG the chain is diagonalized iteratively.Here too, the Fock space grows exponentially reaching very quickly a size, whichcannot be diagonalized anymore in suitable time. One has to truncate the Fockspace selecting a smaller number of states, which will build up the next iteration.This is done in a way best fitting the aim of the procedure. In the NRG thestates with the lowest energies were selected to create the next Hamiltonian.The reason for this was the exponentially decreasing hopping parameters andthe focus on the low energy eigenstates of the problem. In the DMRG thehopping parameters can have arbitrary values. Selecting the lowest eigenstatesin smaller blocks for building up larger blocks fails because of wrong boundaryconditions [119]. In DMRG the new states for an enlarged block are calculatedusing the ground state of the whole system. From the ground state one canbuild up the reduced density matrix [120] for the enlarged block. Let me assumethe whole system is build up from four blocks, see figure 3.4. The middle blocksB2 and B3 correspond to single sites containing only a very limited number of

3.4 Density Matrix Renormalization Group 51

Sta

tes

B1 B2 B3 B4

|i1〉 |i2〉 |i3〉 |i4〉

|i〉 = |i1〉 ⊗ |i2〉 |k〉 = |i3〉 ⊗ |i4〉

Figure 3.4: One iteration step in DMRG. The blocks are named B1, B2, B3 and B4with states |i1〉, |i2〉, |i3〉 and |i4〉. A basis set is formed by tensor product for the twoleft blocks and two right blocks, respectively.

states. The remaining blocks B1 and B4 represent all sites of the left or theright part. Both are supposed to already contain a truncated Fock space. I willnow explain the method how to build up the next enlarged block.Each block consists of a finite number of states |ik〉. From these states theHamiltonian for the whole system can be set up. For this Hamiltonian onesearches the ground state by the Lanczos method or some similar technique[121,122]. Assuming the chain is represented by 4 blocks the ground state canbe written as

Ψ =∑

i1,i2,i3,i4

Φi1,i2,i3,i4 |i1〉 ⊗ |i2〉 ⊗ |i3〉 ⊗ |i4〉.

One can now combine both left blocks forming one new block |i〉 = |i1〉 ⊗ |i2〉.The same is done for the two right blocks. As a result, one can write the groundstate in the states of the two big blocks

Ψ =∑

i,k

ψi,k|i〉 ⊗ |k〉.

From the coefficients ψi,k one can now form the reduced density matrix for theleft blocks

ρi,j =∑

k

ψi,kψj,k, (3.13)

where i, j are indices of the left block, k an index of the right block and ψi,k isassumed to be real. This matrix ρi,j is supposed to have a maximal dimensionof about dim ρij ≈ 1000, as one has to diagonalize this matrix completely ineach iteration.Let me assume that I want to calculate the expectation value of a bounded

operator ‖A‖ = maxφ

∣

∣

∣

〈φ|A|φ〉〈φ|φ〉

∣

∣

∣ = cA acting only in the left enlarged block.

This expectation value is given by the trace of the operator times the density


operator ρ. As the operator acts only at the left block, the density operatorcorresponds to equation (3.13). As ρ is a symmetric matrix one can write itas ρ =

∑

i |λi〉λi〈λi|, where the eigenvalues are supposed to be ordered fromlarge to small. One wants to truncate the dimension of the block from dim ρij

to m < dim ρij, thus taking the eigenstates with the largest eigenvalues. Theexpectation value can be written as

〈A〉 = Tr ρA

≈m∑

i=0

λi〈λi|A|λi〉.

The error of the truncation for this operator can be easily approximated by∣

∣

∣

∣

∣

〈A〉 −m∑

i=0

λi〈λi|A|λi〉∣

∣

∣

∣

∣

=

dim∑

i=m+1

λicA

being proportional to the sum of the neglected smallest eigenvalues. Thus theeigenstates with the largest eigenvalues of the density operator best describethe operator A [113]. This holds for every bounded operator, especially for thepart of the Hamiltonian describing the system of the two blocks. One can alsoargue that these states best describe the ground state wave function [123] ormaximize the quantum entanglement [124].The algorithm now works as follows: First one calculates the ground stateof four single sites. From this a combined block including two single sites iscreated. So the new system consist of the combined block and three single sitesfrom which a combined block of three sites is created. Going on like this thewhole chain can be built up. After setting up the whole chain, one can usethe same algorithm to improve the basis stored in each combined block by firstenlarging the left blocks and later the right blocks. Iterating this procedureone can get a very accurate ground state wave function of an one-dimensionalchain. With this wave function one can calculate static expectation values orcorrelation functions of the chain. Of course, this includes expectation valuesof the impurity itself.

3.4.2 Calculation of dynamical properties

The DMRG focuses on the ground state of the system. One usually is notable to calculate more than a very few excited states. Therefore, one is notable to use Lehmann’s formula for calculating spectral functions. The easiestway calculating a spectral function is using a Lanczos expansion of the spectralfunction [125, 126] calculating the first few moments. But this expansion failsfor more complex dynamical spectral functions.An alternative approach is the correction-vector method [126, 127]. A veryelegant way of formulating it, also improving its accuracy, was introduced byE. Jeckelmann [128]. The Laplace transformed retarded Green’s function (3.10)for T = 0 can be written as

GAB(ζ) =⟨

Ψ∣

∣

∣A

1

ζ −HB∣

∣

∣Ψ⟩

,

3.4 Density Matrix Renormalization Group 53

where Ψ is the ground state of the chain. For numerical calculation it is neces-sary that ζ = ω + iη has positive finite imaginary part η. Instead of invertingthe operator (ω + iη −H) one sets up the following equation for the state |Y 〉

((E0 + ω −H)2 + η2)|Y 〉 = −ηB|Ψ〉, (3.14)

where E0 is the ground state energy of the chain. From this one can calculatethe spectral function by

ρ(ω) = − 1

πIm〈Ψ|A|Y 〉.

One can now formulate this linear equation as a minimization problem [128] ofthe following functional

W (ω, η) = 〈Φ|(E0 + ω −H)2 + η2|Φ〉 + 2η〈Φ|B|Ψ〉.

The state |Φ〉, which minimizes this functional equals the solution to the equa-tion (3.14)

|Φmin〉 = |Y 〉.With this approach it is possible to calculate the spectral function of the An-derson model by using the DMRG.

3.4.3 Algorithm

At the end of this section I will summarize the DMRG algorithm. First, onebrings the Anderson impurity Hamiltonian into the discrete form of a linearchain. In contrast to the NRG it is possible to discretize the conduction band,for example, in linear intervals resulting in non-exponentially decreasing hop-ping parameters. By this one can control the resolution of the conduction band.As usual, the only interaction between spin-up and spin-down electrons occuron the impurity. Thus it is possible to build up the chain with all the up-electrons to the left, the impurity sites in the middle and the down-electrons tothe right. This results in a small local Fock space consisting of only two states:no electron or one electron. This accelerates the whole procedure. One nowuses the DMRG for finding the ground state |Ψ〉 of the whole chain. By increas-ing and decreasing the size of the outer combined blocks, called sweeping, oneimproves the basis set used for setting up the Hamiltonian and finally is ableto calculate a very accurate ground state wave function. With |Ψ〉 it is possibleto compute static expectation values, such as the occupation of the impurity.When calculating the spectral function one cannot simply use the basis set justgained for the ground state. This will give a very poor result for the spectralfunction as the basis set is optimized for the ground state and cannot describethe exited state B|Ψ〉 very well. For each frequency ω, one has to calculate theexited state B|Ψ〉 and the state |Y 〉 as defined in equation (3.14). One has toimprove the basis set by sweeping for these two states and the real part of thespectral function |X〉 given by

|X〉 =H − E0 − ω

η|Y 〉.


This means that one includes not only the ground state wave function into thereduced density matrix but also these three exited states [128]. The densitymatrix is then simply written as sum

ρij =∑

p

αp

∑

k

ΨαikΨ

αjk,

where p labels the four states and αp is some amplitude for the importance ofthe state. In my calculations I usually used α = 0.4 for the ground state andα = 0.2 for each of the exited states. By sweeping for each frequency point oneimproves the basis set and is finally able to calculate the spectral function.The coefficient η results in a Lorentzian broadening of the spectral function.Decreasing η results in a worse convergence to the solution of the equation forthe correction-vector (3.14). Not only more time is needed for finding the solu-tion, one also has to include more states m in the density matrix for calculatinga stable solution.

CHAPTER 4

NRG and DMRG Spectral Functions

for the Impurity Model

4.1 Introduction

This chapter deals with a comparison between the numerical renormalizationgroup (NRG) and the density matrix renormalization group (DMRG) as im-purity solvers. Both methods were introduced in chapter 3. The NRG is wellestablished for calculating spectral functions for one- or two-orbital Andersonmodels [94, 105]. These kind of calculations have been done for approximately20 years. In contrast, the DMRG algorithm, which was used here for the com-putation of spectral functions, is rather young existing in this form for only 10years [126–128]. Calculations for impurity systems have been performed for 5years [115, 116]. From this fact it is clear that one should test the DMRG asimpurity solver before using it in the framework of the DMFT. Intensive testswere performed in the Ph.D. thesis of C. Raas [129].As the DMRG spectral functions are Lorentz-convoluted, it is recommendableto look at the deconvolution of the data. Therefore, I will give a short introduc-tion into the topic of deconvolution in this chapter. Finally, I will compare thespectral functions of the DMRG and the NRG for different interaction strengths,doping and magnetic fields. All three parameters are important, when usingthe DMRG in the framework of the dynamical mean field theory for calculatingmagnetic phase diagrams. Based on these results I will discuss the strengthsand the weaknesses of the DMRG as an impurity solver.

56 NRG and DMRG Spectral Functions for the Impurity Model

-2 -1 0 1 2ω

0

0.5

1

1.5

2

2.5

3

ρ(ω

)

U/Γ=0U/Γ=2.8U/Γ=5.6U/Γ=8.4U/Γ=11.2U/Γ=14.1increasing U

Figure 4.1: Spectral functions from DMRG calculations for the SIAM for Γ = 0.07 anddifferent interaction strengths. Friedel sum rule states ρ(0) = 1

πΓ≈ 4.54 in this case.

Within the DMRG a broadening of η = 0.05 was used.

The single impurity Anderson model (SIAM), as introduced in chapter 3, reads

HAnd =∑

σ

∫ D

−Ddǫ ǫc†ǫ,σcǫ,σ +

∑

σ


†d,↓cd,↓

∑

σ

∫ D

−DdǫV (ǫ)

√

ρ(ǫ)(c†ǫ,σcd,σ + c†d,σcǫ,σ).

A constant coupling V (ǫ)√

ρ(ǫ) = const between the conduction band and theimpurity is chosen throughout this chapter. The hybridization is defined as

Γ = πV (ǫ)2ρ(ǫ) = πV 2ρ for ǫ ∈ [−D,D].

Figure 4.1 shows spectral functions of the SIAM for different interaction strengthsobtained by the DMRG. The general form of the spectral functions agrees verywell with the known results [98]. One can see the resonance at the Fermi en-ergy, ω = 0, called Kondo resonance. For large enough interaction strengths,additional high energy peaks, called Hubbard peaks, are visible. The spectralweight is transferred from the central peak to the Hubbard peaks when increas-ing the interaction. But already in this figure one should notice that the Friedelsum rule [98], which states that the half-filled SIAM spectral function is pinnedat ω = 0 to ρ(0) = 1

πΓ , is not fulfilled. I will discuss the spectral functions inmore detail later in this chapter.

4.2 Used Discretization Schemes 57

-1.5 -1 -0.5 0 0.5 1 1.5ω

0

0.1

0.2

0.3

0.4

0.5

ρ(ω

)

-0.5 0 0.5ω

0

0.5

1

1.5

2

2.5

3

ρ(ω

)

U=0 exact chainU=0 DMRGU=0 exact

-0.5 0 0.50.48

0.485

0.49

ρLin

ρLog

Figure 4.2: Left: Bath spectral function obtained by the diagonalization of the hoppingchain. Two different discretization schemes were used. The black (red) line correspondsto linear (linear and logarithmic) intervals. The broadening η = 0.05 was used. Right:U = 0, Γ = 0.07 spectral function of the SIAM. The black points and the red lineare obtained by diagonalizing the hopping chain. The green curve represents the exactsolution to the SIAM. For the DMRG calculations a discretization into 81 intervals wasused.

4.2 Used Discretization Schemes

Before using either the NRG or the DMRG for the SIAM, one has to discretizethe band states (chapter 3). The discretization for the DMRG is quite arbitrary.In figure 4.2 I show two resulting bath Green’s functions and the non-interactingimpurity spectral function. The bath spectral function represents the form ofthe bath, which couples to the impurity. The results throughout this chapterwere done for a constant bath spectral function ρ0(ǫ) = 0.5 for ǫ = [−1, 1]. Ihave tested two kinds of discretization schemes in the DMRG. The black linein the left panel of figure 4.2 corresponds to a pure linear discretization, whilethe red line corresponds to a linear discretization for large frequencies |ω| > 0.1and a logarithmic discretization for |ω| < 0.1. The reason for the logarithmicpart is to improve the resolution around the Fermi energy. The bath spectralfunctions were calculated by exact diagonalization of the chain by

ρ(ω + iη) = − 1

πIm

⟨

Ψ0

∣

∣

∣

∣

c1

ω + iη −Hc†∣

∣

∣

∣

Ψ0

⟩

, (4.1)

where c† creates an electron on the last site of the bath sites, which directlycouples to the impurity. I have used the same broadening η in this calculation asin the DMRG calculations. One can see, that this approach quite nicely repro-duces a nearly constant bath continuum near ω ≈ 0. The discrepancy between


0.5 and 0.48 as well as the curvature at the band edges can be explained by thebroadening. The logarithmic broadening shows slight oscillations around thepoint, where I changed from linear to logarithmic discretization. Throughoutthis chapter I used 81 intervals for the discretization of the bath continuum.The complete DMRG chain with splitting of the spin-up and the spin-downstates to the left and right of the impurity adds up to 164 sites. The rightpanel in figure 4.2 corresponds to the non-interacting SIAM, U = 0. The redline and the black points, which lie exactly on the red line, are the calculatedand broadened spectral function in discretized form. “Exact” means that thehopping chain was diagonalized without DMRG. The green curve correspondsto the exact spectral function of the SIAM. The DMRG reproduces the exactspectral function, which can be obtained after the discretization of the bathstates. The difference between the exact curve and the discretized spectralfunction is due to the discretization and the broadening.

4.3 Deconvolution

Due to numerical reasons and finite size effects, see chapter 3, it is not possibleto use equation 4.1 for arbitrary small broadening η within the DMRG. From

G(z) =

⟨

Ψ0

∣

∣

∣

∣

c1

z −Hc†∣

∣

∣

∣

Ψ0

⟩

=

∫

ρ(ω)

z − ωdω

⇒ ρ(ω + iη) = − 1

πIm

∫

ρ(ω′)ω + iη − ω′dω

′

=η

π

∫

ρ(ω′)(ω − ω′)2 + η2

dω′ (4.2)

one can see, that the spectral function calculated within the DMRG is con-voluted with a Lorentzian. Deconvolution is an ill-posed problem due to tworeasons. The first reason is a physical one: The unbroadened spectral functionof a finite system, like the finite chain used within DMRG, consists of a finitenumber of delta-peaks. Of course, this number is rather big. Nevertheless,finite size effects will most likely be visible. As one is interested in the spectralfunction, when there is a continuum of bath states, these finite size effects areundesirable. Therefore, it is actually not the goal to do a complete deconvo-lution. But even if a complete deconvolution would be the goal, it would benumerically impossible. The mathematical reason for this is the sensitivity onthe input data points. Only small changes in ρ(ω + iη) can lead to rather bigchanges in ρ(ω). As the inversion of the integral is done numerically, errorswill occur. Another way to formulate the last statement is, that within a givenerror for the inversion there is more than one solution.

4.3.1 Matrix Inversion

A straight forward idea for the deconvolution is to consider the spectral functionas a discrete set of points. Thus the integral in equation (4.2) can be written as

4.3 Deconvolution 59

a matrix operation. Let ρi = ρ(ωi) with ωi+1 − ωi = dǫ, one can approximateequation (4.2) as

ρ(ωi + iη) =η

π

∑

j

ρj

(ωi − ωj)2 + η2dǫ =: Lijρj,

defining the matrix of the Lorentz-convolution Lij. This matrix can in principlebe inverted. Thus the desired spectral function can be calculated by

ρj = L−1ij ρ(ωi + iη).

If one uses a fine mesh in frequency space, the matrix Lij will have eigenvalues,which are approximately zero thus making an inversion practically impossi-ble. Pseudo-inverse techniques can be applied yielding results for the spectralfunction. These deconvoluted spectral functions have commonly two problems:They are not positive definite and they show strong oscillations. Thus bettermethods should be used for deconvolution [129].

4.3.2 Maximum Entropy Ansatz

A method, which is frequently used for analytic continuation of Quantum MonteCarlo (QMC) data or image restoration, for example, in astronomy, is MaximumEntropy [130–132]. This kind of analytic continuation is closely related todeconvolution as both methods try to invert the same integral. But in theanalytic continuation of QMC data one tries to calculate ρ(ω) from G(iω).The Shannon information entropy is written as

S[ρ(ω)] = −∫

dωρ(ω) logρ(ω)

D(ω),

in which D(ω) is the default model. The entropy without any further conditionsis maximal for ρ(ω) = D(ω). Throughout this thesis I use a constant defaultmodel D(ω) = 1. The same default model was used in the deconvolution ofDMRG data in the work of C. Raas [129] and P. Dargel [133]. The reasonfor using a constant default model is the wish for a least biased deconvolutionstrategy. Finding the maximum is done under the condition that ρ(ω) is con-sistent with the N calculated data points ρ(ωi + iη), which are supposed to bedeconvoluted, meaning

ρ(ωi + iη) =η

π

∫

ρ(ω)

(ω − ωi)2 + η2dω. (4.3)

Using Lagrange-multipliers λi to enforce this condition, one can find the ex-tremum of the entropy by

0 =δS[ρ(ω)]

δρ(ω)

=

∫

dω

(

− log ρ(ω) − 1 +∑

i

λiη

π

1

(ω − ωi)2 + η2

)


giving an ansatz for the spectral function as

ρ(ω) = exp

(

∑

i

λiη

π

1

(ω − ωi)2 + η2

)

.

This ansatz includes N free parameters λi, which can be calculated by equa-tion (4.3). Thus one has to solve N non-linear coupled equations yielding theparameters λi. For the results shown in this chapter the system of equationswas solved by steepest descent until the desired precision of about

1

Nχ2[ρ(ω)] :=

1

N

∑

i

(

ρ(ωi + iη) − η

π

∫

ρ(ω)

(ω − ωi)2 + η2dω

)2

< 0.0002

was gained.The algorithm can be improved by additional assumptions. For example, onecan enforce the norm

∫

dωρ(ω) = 1 being always fulfilled. Another point is toassume error afflicted data. Then one can include the variance

χ2[ρ(ω)] =∑

i

(

ρ(ωi + iη) − η

π

∫

ρ(ω)

(ω − ωi)2 + η2dω

)2

into the functional, which is maximized, yielding

Q = S[ρ(ω)] −Aχ2[ρ(ω)].

The constant A can then be used for tuning the resulting ρ(ω), so that χ2 equalsthe magnitude of the assumed errors.Figure 4.3 shows an example for a deconvolution. I assume no errors in the pureDMRG data. The black line corresponds to the DMRG output for a SIAM withΓ = 0.07 and U = 0.8. The blue line represents the NRG spectral function.One should notice two things: The Kondo-peak is strongly smeared out inthe DMRG-data. On the other hand, the Hubbard satellites are much betterresolved within the DMRG than in the NRG. Using now deconvolution on theDMRG data, all peaks become stronger. Nevertheless the Kondo peak having awidth of TK ≈ 0.008 cannot be resolved. Moreover, in the deconvoluted spectralfunction additional oscillations occur, as can be seen at ω ≈ ±0.15. Theseoscillations are unphysical and due to the deconvolution. Similar oscillationsoccur in much stronger form in the matrix based deconvolution. It is usuallyvery difficult to get rid of these artifacts. Therefore, it is very important to haveknowledge about this fact, so that one can interpret this oscillations properly.

4.4 Comparison between DMRG and NRG

4.4.1 Single Impurity Anderson Model

I will now present several results which show spectral functions for the SIAM.In all figures I have used Γ = 0.07. The initial DMRG broadening was alwaysη = 0.05. Figure 4.4 shows the spectral function for different ratios between the

4.4 Comparison between DMRG and NRG 61

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1ω

0

1

2

3

ρ(ω

)

DMRG Outputdeconvoluted DMRGCFS NRG

-0.04 0 0.04ω

0

2

4

ρ(ω

)

Figure 4.3: Spectral functions for U = 0.8 and Γ = 0.07. One can see the pure DMRGoutput and the comparison between the complete Fock space (CFS) NRG and thedeconvoluted DMRG spectral function. The initial DMRG broadening was η = 0.05.

interaction strength and the hybridization Γ. Continuous lines correspond todeconvoluted DMRG results and the dashed lines correspond to NRG results.The lower panel in figure 4.4 shows a magnification of the frequencies around theFermi energy ω = 0. Two things should be obvious: The DMRG results resolvethe Hubbard bands at large frequencies very well. But the Kondo resonanceat ω = 0 is resolved very badly. For the NRG results the opposite statementis true. Due to the logarithmic broadening in the NRG, the Hubbard bandsare completely smeared out. Both facts hold true for all figures shown in thischapter. Quite recently it was possible to increase the resolution of the NRGby a special kind of z-averaging [134]. Figure 4.5 shows the spectral functionsfor U/Γ = 11.2. There are two DMRG calculations included correspondingto a linear discretization and a logarithmic discretization. The height of theKondo peak within the logarithmic discretization is reduced, thus failing toimprove the resolution of the low energy features. This is not astonishing, asthe height is mainly limited by the broadening η. Additionally, I have includedin figure 4.5 the NRG spectral functions using the ordinary NRG broadening(CFS NRG) and the high resolution broadening (HR NRG). One can see howthe high resolution NRG can resolve the Kondo peak as well as the Hubbardbands with very high accuracy.For calculating magnetic phase diagrams within the DMFT using the DMRG asan impurity solver, it is necessary that the DMRG is able to calculate spectralfunctions away from half-filling and is able to handle magnetic fields. Figures4.6 and 4.7 show that both cases can be treated using the DMRG. Figure 4.6shows the comparison between the DMRG and the NRG for different on-site


-1 -0.5 0 0.5 1ω

00.5

11.5

22.5

3

ρ(ω

)

U/Γ=8.4U/Γ=11.2U/Γ=14.1

-0.1 -0.05 0 0.05 0.1ω

0

1

2

3

ρ(ω

)

Figure 4.4: Spectral functions for Γ = 0.07 and different interaction strengths. Contin-ues lines correspond to the DMRG, dashed lines to the NRG. The lower panel shows amagnification of the upper panel around the Fermi energy ω = 0.

-1 -0.5 0 0.5 1ω

0

0.5

1

1.5

2

ρ(ω

)

DMRG linDMRG logHR NRGCFS NRG

-0.05 0 0.05

0.5

1

1.5

DMRG linDMRG logHR NRGCFS NRG

Figure 4.5: Spectral functions for Γ = 0.07 for U/Γ = 11.2. The black and thered line correspond to deconvoluted DMRG results with either linear discretization orlogarithmic discretization of the conduction band. The green line represents the highresolution NRG explained in the text (This data was provided by Rok Zitko). The blueline corresponds to the complete Fock space NRG. The inset shows a magnification ofthe region around the Fermi energy.


0

0.5

1

1.5

2

ρ(ω

)

-2 -1 0 1ω

0

0.5

1

1.5

2

ρ(ω

)

NRGDMRG

-2 -1 0 1 2ω

NRGDMRG

εd=-0.3 ε

d=-0.2

εd=-0.1 ε

d=-0.0

Figure 4.6: Spectral functions for the SIAM with Γ = 0.7, U = 0.8, Tk ≈ 0.008, anddifferent on-site energies.

00.5

11.5

22.5

3NRG MinorityNRG MajorityDMRG MinorityDMRG Majority

-1 -0.5 0 0.5 10

0.5

1

1.5

B/Γ=0.007

B/Γ=0.07

Figure 4.7: Spectral functions for the SIAM with Γ = 0.7, U = 0.8, Tk ≈ 0.008, anddifferent magnetic fields.


energies, resulting in different occupations of the impurity. Within the NRGresults one can see, how the spectral weight is shifted to the right, but onecannot see any substructures in the Hubbard bands. In contrast, within theDMRG results one can nicely see, how the left Hubbard band moves into theKondo resonance.Figure 4.7 depicts the results for two different strengths of the magnetic field. Inthe beginning of the NRG, there was the problem that the NRG dramaticallyunderestimated the effect of small magnetic fields. This problem was firstlysolved by introducing the reduced density matrix [111]. The reduced densitymatrix ensures the correct influence of the ground state on the high energyfeatures in the spectral function. As in the DMRG the correct ground stateis used for each frequency, such problems are supposed to be absent. Thisstatement is proven in figure 4.7. The splitting of the majority and minorityspin spectral functions is comparable in strength for both methods. Thus theDMRG is able to handle small and large magnetic fields acting on the impurity.

4.4.2 Two-Orbital Anderson Model

-1.5 -1 -0.5 0 0.5 1 1.5ω

0

0.5

1

1.5

2

2.5

ρ(ω

)

Γ=0.125Γ=0.196

Figure 4.8: Two-orbital Anderson model for U = 0.4, J = 0, U ′ = 0.4, and two differenthybridizations Γ. Continuous lines are DMRG results, dashed lines NRG results.

Finally, it is an interesting question to see, if the DMRG can handle the multi-orbital Anderson model. An intensive study for a simplified two impurity An-derson model was done by S. Nishimoto et al. [135]. Figure 4.8 shows the resultsfor the two-orbital Anderson model. The construction of the one-dimensionalchain was such, that the Fock space states of one orbital were to the left ofthe impurity and the states of the other orbital to the right. Figure 4.8 and4.9 shows that it is possible to do multi-orbital impurity calculations using theDMRG. Interestingly, the deconvoluted DMRG spectral functions in figure 4.8


-1 -0.5 0 0.5 1ω

0

0.5

1

1.5

2

2.5

3

ρ(ω

)

DMRGNRG

-1 -0.5 0 0.5 1ω

0

0.5

1

1.5

2

2.5

3

ρ(ω

)

DMRGNRG

Figure 4.9: Two-orbital Anderson model for U = 0.4, J = 0.1, U ′ = 0.2, and twodifferent hybridizations Γ = 0.125 (left) and Γ = 0.031 (right).

with J = 0 have more spectral weight at the Fermi energy ω = 0 than theNRG. This is in contrast to the calculations in the one-orbital Anderson model.As the Friedel sum rule is supposed to hold in this case, one can see, thatthe DMRG fulfills the sum rule more accurately. For Γ = 0.125 the sum rulepredicts ρ(0) = 2.54, and for Γ = 0.196 it predicts ρ(0) = 1.62. Two-orbitalcalculations using the NRG have the big problem, that the Fock space mustbe built up from 16 states at the same time. Although the shown NRG re-sults were done with Nk = 5000 states kept after the truncation, this numbercorresponds to NK = 70 states in the one-orbital model. This explains thediscrepancies of the NRG results to the Friedel sum rule. Another aspect in thespectral functions are the band edges at ω = ±1. They are again completelysmeared out in the NRG due to the large broadening at large frequencies. TheDMRG spectral function nicely resolve these band edges. Finally, figure 4.9shows results for J = 0.1. Finite Hund’s coupling has a profound consequenceon the spectral function. The Kondo temperature can be considerably reducedby it [52]. The left panel in figure 4.9 shows the spectral function for relativelylarge hybridization Γ, for which no Hubbard peaks are present. In this situationthe NRG and the DMRG are very similar. In the right panel one can see thespectral functions for a small hybridization Γ. The Kondo temperature is verysmall. It can still be resolved by the NRG, but the DMRG fails. In the DMRGspectral function one can see additional substructures in the Hubbard bands.But it is very difficult to say, if these structures are real or effects of the de-convolution. One should perform additional calculations with other conductionband discretization to check this result.


4.5 Summary

In this chapter I compared results for the spectral functions of the NRG andthe DMRG. As the NRG is well established for calculating spectral functionsfor arbitrary interaction strengths, impurity occupation, and magnetic fields,I mainly tested the DMRG algorithm in this chapter. I have shown that theDMRG is able to calculate dynamical properties of a SIAM reliably. The mainproblem for the DMRG is the broadening, which can be comparable to thewidth of the Kondo resonance, thus eventually failing to correctly resolve thelow energy features. On the other hand, the DMRG resolves the high energyfeatures like the Hubbard bands very well, at least better than within the NRG.Of course, one should mention that also in the NRG there are ways to improvethe resolution of the Hubbard bands [134]. A way to improve the resolutionwithin the DMRG is deconvolution. Although possible, there are limits to whatextend structures can be resolved. As the focus of this thesis lies on magneticphases, which are mainly stabilized due to the high energy features, there is thehope, that the DMRG can deal with these situations.One very big advantage of the DMRG to the NRG could be seen in the two-orbital Anderson model. As the discretization within the DMRG and also theconstruction of the chain is rather arbitrary, one has the possibility to split thelocal Fock space states in the DMRG. This leads to a reduction of the numberof states during one iterative diagonalization. In the DMRG, it is possibleto separate the band states for each spin and orbital direction, as they areonly interacting with each other directly on the impurity. This enables one toperform calculations for even more than two orbitals, as the chain size increasesonly linearly with the number of orbitals.

CHAPTER 5

One-orbital Hubbard model

5.1 Introduction

In this chapter I will present my results for the situation, when there is effec-tively only one orbital left at the Fermi level. This can occur in the case offurther symmetry reduction from cubic perovskite to tetragonal structure [14],see chapter 1.2. Assuming that there is no Hund’s coupling between the par-tially filled orbital and the remaining localized electrons, or the remaining bandsare completely filled or empty like in the case of cuprates [14], this situationcan be modeled by the one-orbital Hubbard model (chapter 1.3)

H =∑

ij,σ

tijc†i,σcj,σ − µ

∑

i,σ

ni,σ + U∑

i

ni,↑ni,↓,

where tij describes the hopping of the electrons, µ is the chemical potential, andU the amplitude for the electron-electron interaction. The model was alreadyintroduced in the first chapter. In this chapter I will examine the effects ofall three parameters: Changing the form of the hopping, the ratio betweenhopping and interaction strength, and the occupation of the system by tuningthe chemical potential µ.All calculations in this chapter were performed within DMFT for a Bethe lattice.In the second section I will address the Hubbard model with nearest neighbor(NN) hopping only, resulting in a semi-elliptic DOS. In the third section Icompare DMRG with NRG results for this model. Finally, in the last sectionof this chapter I will introduce an additional next nearest neighbor (NNN)hopping t2 6= 0 leading to an asymmetric DOS. As shown in chapter 2.5, thesemi-elliptic density of states for NN-hopping becomes asymmetric and finallydevelops a singularity at the lower band edge for t2 > 0 and t2/t1 > 1/4. Fort2 < 0 the DOS is reflected at ω = 0 compared to t2 > 0. All the results remain

68 One-orbital Hubbard model

-2 -1 0 1 2ω/W

0

1

0

1

ρ(ω

)W

0

1

0

1

2

1.1 1.2 1.3 1.4 1.5U/W

0

0.005

0.01

0.015

T/W

U/W=1

U/W=1.5

U/W=1.505

U/W=1.6

Metal Insulator

Figure 5.1: Metal insulator transition within DMFT for a Bethe lattice with semi-elliptic DOS. The left side of the figure shows the evolution of the spectral functionρ(ω) with increasing interaction strengths U/W = (1, 1.5, 1.505, 1.6) for T = 0. Theright side shows the first order transition lines from a metal to an insulator (right line)and from an insulator to a metal (left line). At T/W ≈ 0.01 there is a critical point,above which there is only a cross-over between metallic and insulating solutions.

valid with occupation n = 1 − x −→ n = 1 + x for t2 −→ −t2. Therefore I willassume through out this chapter t2 ≥ 0.Most of the results in this chapter are published in Physical Review B [136] andNew Journal of Physics [137].

5.2 Semi-elliptic Density of States

5.2.1 Metal Insulator Transition

Assuming that there is only NN-hopping t, the Bethe lattice has a symmet-ric semi-elliptic DOS, see chapter 2.5. The bandwidth W = 4t of the non-interacting system is in the following used as the energy unit. The calculationsare performed with DMFT/NRG for a half-filled system n = 1 using the NRGdiscretization parameter Λ = 2, and NK = 1000 kept states (see chapter 3).As I will show now, the paramagnetic metal insulator transition (PMIT) isstrongly connected to the magnetic phase diagram of the Hubbard model. ThePMIT is a numerically well analyzed phenomenon within DMFT [106,138–141].The results obtained in my own calculations can be seen in figure 5.1 and aresummarized as follows: When approaching the PMIT from the metallic phasea three peak structure develops with two Hubbard bands and a quasi-particlepeak at the Fermi-energy ω = 0 [106,138]. The width of the quasi-particle peak

5.2 Semi-elliptic Density of States 69

decreases for increasing interaction strengths. Finally at a critical interactionstrength the quasi-particle peak vanishes. Thus an insulator is formed. ThePMIT occurs for an interaction strength of about U/W ≈ 1.5 at T = 0. The leftpanel of figure 5.1 shows the evolution of the spectral function for increasinginteraction strength U for T/W = 2 · 10−4. When starting the DMFT self-consistency with an insulating medium, obtained from calculations with largeinteraction strength, the transition occurs at a lower critical value of the interac-tion, at which the gap vanishes. The right side of figure 5.1 shows these PMITtransition lines. The right line shows the transition starting with a metallicself-energy, while the left line corresponds to the transition when starting withan insulating self-energy. Between both lines there is a hysteresis region, whereboth insulating and metallic solutions can be stabilized. Therefore, the transi-tion is of first order for temperatures below the critical temperature. Above thecritical temperature there is a smooth cross-over between the metallic and theinsulating phase. As I did not perform any Λ → 1 scaling in the NRG, the tran-sition lines and the critical endpoint do not exactly lie upon the DMFT/NRGresults by R. Bulla et al. [106,138], but they are very close to them.

5.2.2 Magnetic Phase Diagram

The Hubbard model at half-filling exhibits super exchange, as described inchapter 1.4. Therefore it is not surprising to find a strong tendency towardsantiferromagnetism. The antiferromagnetic phase in the Hubbard model athalf-filling is a well analyzed fact, too [142–145]. It can be shown by per-turbation theory that for an unfrustrated lattice, t2 = 0, an antiferromag-netic phase exists at half-filling and T = 0 for any finite interaction strengthU > 0 [142, 143]. The paramagnetic metal insulator transition, shown in fig-ure 5.1, has a smaller critical temperature than the Neel-temperature of theantiferromagnetic phase [70, 140, 146, 147]. This means that the PMIT is com-pletely covered by the antiferromagnetic phase at half-filling. For analyzing thePMIT in this model the antiferromagnetic phase has to be suppressed artifi-cially during the DMFT self-consistency cycle by, for example, averaging thespin components of the effective medium. A phase diagram, as seen in V2O3

(chapter 1.2.1), cannot be seen in the Hubbard model with semi-elliptic DOS.It has been argued that frustration might be able to suppress the antiferromag-netic phase revealing the PMIT [70, 147, 148]. I will come back to this pointlater in this chapter.Besides antiferromagnetism, Nagaoka ferromagnetism was found in the Hub-bard model [144, 149–151] for a simple cubic lattice. Interestingly, it is impos-sible to stabilize this kind of ferromagnetic phase for a Bethe lattice. In therigorous derivation of Nagaoka for this ferromagnetic state [149], closed loopsin the lattice structure are essential. As the Bethe lattice does not possess anyclosed loops this absence of Nagaoka ferromagnetism is in complete agreementwith the theory. The interesting fact is, that the lattice structure only entersthe DMFT calculation via the DOS. So changing the DOS from Gaussian withexponentially “long tails” for the hyper-cubic lattice, where Nagaoka ferromag-netism has been observed, to the semi-elliptic DOS with finite support, averts


Figure 5.2: Antiferromagnetic polarization for a semi-elliptic DOS and T/W = 2 ·10−4.The phase diagram was calculated with metallic starting medium in the upper panel(insulating starting medium in the lower panel). The antiferromagnetic Neel-phase(yellow color) exists only exactly at half-filling. Black is the paramagnetic phase awayfrom half-filling and the white part the incommensurate spin density wave (IC). Thehorizontal black lines correspond to the critical values of U/W = 1.5 (upper panel)and U/W ≈ 1.2 (lower panel) of the PMIT. In the lower panel the transition lines ofthe calculation with metallic starting medium are included. The phase diagrams werecreated by fitting of approximately 200 data points distributed in each phase diagram.


the existence of this ferromagnetic state. This stresses the influence of thelattice on the DMFT calculations.The phase diagram for very low temperature T/W = 2 · 10−4, different inter-action strengths and chemical potentials can be seen in figure 5.2. Both panelsshow the antiferromagnetic polarization p = n↑ − n↓, which is colored yellowcorresponding to magnitudes larger than p > 0.8. These phase diagrams werecreated by using different starting media for the DMFT calculation. The upperpanel in figure 5.2 corresponds to a metallic starting medium, while the lowerpanel corresponds to an insulating starting medium. The media influence thecritical value of the interacting strength U/W at which the metal insulator tran-sition occurs in the paramagnetic phase (figure 5.1). In both phase diagramsthe antiferromagnetic phase exists only exactly at half-filling. This Neel-stateis pinned to n = 1 and does not change upon slightly changing the chemicalpotential. Looking at the metallic starting medium, the paramagnetic metalinsulator transition would occur for U/W ≈ 1.5, which is now covered by theantiferromagnetic state. Nevertheless, the magnetic phase diagram changes atthis point. For interaction strengths below this value, the paramagnetic phase,represented by the black color, adjoins directly the antiferromagnetic phase athalf-filling. The transition is of first order as both occupation and polarizationjump, as can be seen in figure 5.3. There exists no stable solution for occupa-tions between n ≈ 0.85 and n = 1. This region can be interpreted as phaseseparated between a paramagnetic solution away half-filling and an antiferro-magnetic solution at half-filling [142–144]. The situation completely changesfor interaction strengths above the critical value of the paramagnetic metal in-sulator transition, U/W = 1.5. If one changes the chemical potential µ forU/W > 1.5, one will find a region of parameters where the DMFT calculationsdo not converge, neither to an antiferromagnetic state nor to a paramagneticstate. This regions lies between the paramagnetic phase and the antiferromag-netic phase at half-filling. An example for a DMFT calculation in this regioncan be seen in figure 5.4. Both occupation and polarization oscillate with theDMFT iteration, and no convergent solution can be found. If one uses for exam-ple Broyden mixing [152] during the DMFT self-consistency circle, one can finda meta-stable solution, for the doped antiferromagnetic state. But as one stopsusing the mixing technique, the solution starts to iterate away and soon beginsto oscillate again. This phase is usually interpreted as an incommensurate spindensity wave, as found in these regions earlier by other authors [145,153–156].Changing the chemical potential leads to a phase transition from the Neel-state at half-filling to this incommensurate phase and later into a paramagneticphase. The phase diagram is symmetric towards half-filling, due to the sym-metric DOS. Therefore the same behavior can be seen for electron doping.Although the PMIT is covered by the antiferromagnetic phase, it is an inter-esting observation that it can be seen in the magnetic phase diagram, as atthis point the phase separated region vanishes and the incommensurate mag-netic phase emerges. This shows the close connection between both phenomena.That this transition occurs at the same interaction values is not by chance, canbe seen in the lower panel of figure 5.2. The antiferromagnetic phase diagramdepends on the starting medium, too. The lower panel shows the same calcula-


tions as before, but for an insulating starting medium. The critical interactionstrength for the PMIT lies at U/W ≈ 1.2. The transition in the magneticphase diagram between the phase separated region and the incommensuratephase occurs now at an interaction strength slightly above U/W = 1.2. Thisdiscrepancy between both critical values of the interaction strength can bemost likely explained by a finite resolution effect in the data points. I haveadditionally included the transition lines of the antiferromagnetic and the in-commensurate phase with metallic starting medium into the figure. So one canvery nicely see, how the incommensurate phase starts at smaller interactionstrengths. One should notice, that for weak interaction strengths there is ahysteresis region, where one can obtain either an antiferromagnetic solution athalf-filling or a paramagnetic solution away from half-filling. This is a commonfeature for a first order transition and can be seen in more detail in figure 5.3.The interaction value, at which the transition occurs, depends on the startingmedium.The incommensurate spin density state (IC) exists for temperatures belowT/W ≈ 0.01, see figures 5.6 and 5.5. If there is a close relation between the tran-sition temperature of the IC phase and the critical temperature of the PMIT isdifficult to determine, as the temperature resolution of the data points is badin this region. Interestingly, the transition value between the phase separatedand the IC phase seems to remain at U/W ≈ 1.5 for increasing temperature,while the PMIT interaction value moves towards smaller values of U/W . Whenincreasing the temperature it eventually becomes possible to dope the antifer-romagnetic phase. This can be seen in the color gradient in figures 5.5 and 5.6,signaling a decreasing polarization, at the antiferromagnetic phase boundaries.The reason, why the IC-boundary remains at U/W ≈ 1.5, is most likely dueto this effect. The phase boundaries are smeared out and a doped antiferro-magnetic phase is stabilized by increasing the temperature. For a temperatureT/W = 0.03, figure 5.6, the IC phase has completely vanished and the antifer-romagnetic polarization smoothly decreases from half-filling towards its phaseboundary away from half-filling.

5.2.3 Spin Density Waves

In the last section I have shown that regions of non-convergent DMFT calcula-tions exist in the magnetic phase diagram. I have interpreted these regions asincommensurate spin density waves, as evidence for these were given by otherauthors [145,153–156]. I will show here, that it is in principle possible to stabi-lize such phases for a Bethe lattice with NN-hopping. In the theoretical part onthe Bethe lattice in chapter 2.5, I have used the following equation for derivingthe local Green’s function G00(ζ)

G−100 (ζ) = ζ − t2

∑

i∈NN

G(0)ii (ζ) → ζ − (t⋆)2G00(ζ), (5.1)

where t is the hopping amplitude, z the coordination number of the lattice, andthe scaling t =: t⋆√

zfor z −→ ∞ holds. One assumes a homogeneous lattice,


0.85

0.9

0.95

1

1.05

Occ

upat

ion

0.25 0.26 0.27 0.28 0.29µ/W

0

0.2

0.4

0.6

0.8

Mag

netiz

atio

n

Metallic InputInsulating Input

Figure 5.3: First order transition from the paramagnetic state away from half-filling tothe antiferromagnetic Neel-state at half-filling for T/W = 2 · 10−4, U/W = 1 and twodifferent starting media in the DMFT calculation.

0 10 20 30 40 50DMFT iteration

-0.5

0

0.5

Mag

netiz

atio

n

0.8

0.85

0.9

0.95

Occ

upat

ion

Figure 5.4: Non-convergent DMFT calculation for T/W = 2 · 10−4, U/W = 2, andµ/W = 0.28. The occupation and magnetization oscillate with the iteration number.No convergent state can be found.


Figure 5.5: Magnetic phase diagram for T/W = 0.007. The IC phase still exists forU/W > 1.5. For small interaction strengths the jump of the magnetization at thephase boundary vanishes and there is now a smooth transition.

Figure 5.6: Magnetic phase diagram for T/W = 0.03. The IC phase has completelyvanished.


Figure 5.7: Bethe lattice with coordination number z = 4 and a ABC-structure. EachA-site contacts three B-sites and one C-site.

which leads to G(0)ii (ζ) = G00(ζ). One can now assume a phase with broken

symmetry, in which neighbors have different self-energy.For a bipartite lattice one can immediately write down the equations for an AB-structure, because an A-site has only B-neighbors and vice versa. For longerperiods it is more difficult to find a lattice division supporting, for example,an ABC-structure. But it is possible to define such structures for a Bethelattice, where A-sites have direct contact to almost only B-sites and all B-sites contact almost only C-sites and so on. Almost only means that eachA-site contacts one C-site and z − 1 B-sites (see figure 5.7). From this latticestructure one can now set up the equation for each lattice site using equation(5.1). For a finite coordination number the sum over the nearest neighbors inequation (5.1) has contributions of two different Green’s functions. But in thelimit z −→ ∞ and the proper scaling of the hopping parameter there remainsonly one contribution. So there are N equations for N different local Green’s


0.5

0.55

0.6

0.65

Occ

upat

ion

0 2 4 6 8 10 12 14 16effictive medium

-0.4-0.2

00.20.40.6

Mag

netiz

atio

n

electron magnetizationspin magnetization

Figure 5.8: Stable spin density state away half-filling in the Kondo-lattice model withperiod N = 14. The upper panel shows the occupation for each of the 14 lattice sites.The lower panel shows the magnetization of the electrons and of the spins.

functions, which define a closed set of equations

G−11 (ζ1) = ζ1 − (t⋆)2G2(ζ2)

G−12 (ζ2) = ζ2 − (t⋆)2G3(ζ3)

G−13 (ζ3) = ζ3 − (t⋆)2G4(ζ4)

...

G−1N (ζN ) = ζN − (t⋆)2G1(ζ1).

This can be solved yielding N local Green’s functions from which the new mediafor the next DMFT iteration are calculated. For an AB-structure this procedureyields the same new media as the matrix inversion shown in chapter 2.4.2 aboutDMFT.Unfortunately, one cannot tell from a non-convergent solution, what is thecorrect period stabilizing the solution. Even though one can read off sucha period in figure 5.4, this does not mean that using this will work. Thismeans that one has to try different periods N and hope to find the correctone. For one DMFT iteration for periodicity N , one has to solve N differentAnderson models. This makes long periods a heavy numerical task. By the endof this thesis I did not succeed in stabilizing a spin density wave in the Hubbardmodel for strong interaction. But it was possible in another project to stabilizesuch state in the Kondo-lattice model within DMFT. In this model there is alocal spin S, which is coupled to the electrons via a spin-spin interaction withamplitude J , instead of a local density-density interaction. The model can be

5.3 Comparison of DMRG and NRG 77

written as

H =∑

i,σ

ti,jc†i,σcj,σ − µ

∑

i,σ

ni,σ + J∑

i

~Si · ~si,

where ~s is the spin operator for the electrons at each site. Also in this modelthere is an antiferromagnetic phase at half-filling, which eventually becomesunstable towards a spin-density wave [157] (and references within). Figure 5.8shows an example for a stable spin density wave for J/W = 0.5, µ/W = −0.225,T ≈ 0, and periodicity N = 14. One should notice that the occupation andmagnetization seems to oscillate with a period of N ≈ 3, which however didnot stabilize the system.The example given shows that such order is possible. Nevertheless, the methodused for stabilizing such states cannot be recommended, as the numerical effortis very high. Additionally, such solutions seem to be rather unstable towardsnumerical errors, making it very hard to find them.

5.3 Comparison of DMRG and NRG

In principle the DMRG is an alternative to the NRG as impurity solver for theDMFT [115–118]. For the Bethe lattice it is possible to formulate the DMFTself-consistency equation as [70]

G−1(z) = t2G(z),

in which t is the hopping parameter in the Hubbard model, G(z) the impu-rity Green’s function and G the effective medium of the DMFT. Hence, for theBethe lattice it is not necessary to calculate the self-energy for determining theeffective medium. As it is simpler within the DMRG to calculate the impu-rity Green’s function than calculating the self-energy, I used this formulationof the self-consistency. In the DMRG one usually calculates G(ω + iη). Eventhough the self-consistency can be formulated for arbitrary η, for determiningthe hopping parameters from the effective medium, which are needed in thenext DMRG calculation, one needs G−1(ω + i0). Therefore a deconvolutionmust be performed after every impurity calculation. The PMIT was analyzedwithin the DMFT and DMRG as impurity solver in several works [158–160].They showed that it is possible to use DMRG as impurity solver for DMFTand discussed spectral features close to the Mott transition. I will here concen-trate on magnetic phases. Because the antiferromagnetic phase at half-filling isthe most pronounced magnetic phase in the one-orbital Hubbard model, I willconcentrate on this state.Figure 5.9 shows the antiferromagnetic DMFT results for U/W = 0.5; 0.8;1.0; 1.5 and half-filling. The solutions were calculated separately with theDMRG and the NRG. The DMRG was performed for a bandwidth of W = 0.2and a broadening η = 0.05. The shown DMRG results are already deconvo-luted. For the NRG calculations, I used the same bandwidth and the usualfrequency dependent broadening with b = 0.8, as described in chapter 3.3.3.For U/W = 0.05 (upper left panel) the antiferromagnetic gap is too small to be


-1.5 -1 -0.5 0 0.5 1 1.5ω/W

0

2

4

ρ(ω

)W

-1.5 -1 -0.5 0 0.5 1 1.5ω/W

0

1

2

3

ρ(ω

)W

-1.5 -1 -0.5 0 0.5 1 1.5ω/W

0

1

2

3

ρ(ω

)W

-1.5 -1 -0.5 0 0.5 1 1.5ω/w

0

1

2

3

ρ(ω

)W

NRG majority spinNRG minority spinDMRG majority spinDMRG minortiy spin

U/W=0.5 U/W=0.8

U/W=1 U/W=1.5

Figure 5.9: Antiferromagnetic DMFT calculations performed with DMRG and NRG asimpurity solver for different interaction strengths U/W . The DMRG results are alreadydeconvoluted. For U/W = 0.5 the gap is to small to be resolved by the DMRG. Thusthe DMRG failed to stabilize the antiferromagnetic phase.

resolved by the DMRG. The gap-width for this interaction strength, as calcu-lated by the NRG, is ∆AF ≈ 0.015. With the mentioned Lorentzian broadeningthis cannot be resolved and is too small even after deconvolution. Thus the solu-tion of the DMFT with DMRG for this interaction is a half-filled paramagneticmetal. In contrast, the NRG can resolve this gap resulting in an antiferromag-netic insulator with strong van-Hove singularities at the gap edges. For strongerinteractions DMFT with DMRG can stabilize the correct solution, meaning anantiferromagnetic state. If one looks carefully, one will notice that even if thereis an antiferromagnetic solution for U/W = 0.8 and U/W = 1.0, DMRG failsto really open the gap. The DMRG results would actually predict an antiferro-magnetic metallic state for these intermediate interactions. The reason for thisis again the broadening. The gap-width in both system has the same order ofmagnitude as the broadening η. Only for stronger interactions U/W > 1.5 onecan clearly see a gaped antiferromagnetic state in the DMRG calculations. Theinteraction strengths, at which the DMRG is able to open the gap and form anantiferromagnetic solution, depend on the ratio η/W . Using smaller broaden-ing will make it possible to better resolve the low frequency parts including theantiferromagnetic gap. But using a smaller broadening η will, of course, resultin a heavier numerical task.Another big difference between the spectral functions of the NRG and theDMRG is the structure of the Hubbard bands. As the broadening in the NRG islarge for large frequencies, the Hubbard bands do not show additional structuresfor U/W ≥ 0.8. In the NRG one can only see for U/W = 0.5 van-Hove

5.3 Comparison of DMRG and NRG 79

-1.5 -1 -0.5 0 0.5 1 1.5ω/W

0

0.5

1

1.5

2

ρ(ω

)W

NRGDMRG

-1.5 -1 -0.5 0 0.5 1 1.5ω/W

0

1

2

3

ρ(ω

)W

NRG majority spinNRG minority spinDMRG majorityDMRG minority

Figure 5.10: DMRG and NRG doped spectral function for U/W = 1 and T ≈ 0. Theleft panel shows the paramagnetic solution for µ/W = 0.1. The right panel showsantiferromagnetic solutions for µ/W = 0.3.

singularities and a shoulder in the Hubbard bands. The DMRG results includealways two structures in each Hubbard band. There are theoretical resultson additional structures in the antiferromagnetic state [153, 161–164]. Theseadditional structures are created by virtual movement of electrons or holes inlong paths. If the structures found here are related to this must be furtherinvestigated. It is possible that they are only artifacts of the interplay betweenbroadening and deconvolution. All shown results in figure 5.9 were calculatedusing the same broadening η = 0.05. One should do the same calculations usingdifferent η. If the structures remain at their positions, they are most likely ofphysical origin.Besides the antiferromagnetic behavior for different interaction strengths, oneshould also compare the doping behavior. Figure 5.10 shows the spectral func-tions of antiferromagnetic calculations for two different chemical potentialsµ/W = 0.1 and µ/W = 0.3. For U/W = 1 the system jumps directly froma paramagnetic metal away from half-filling into an antiferromagnetic insulatorat µ/W ≈ 0.27, as I have discussed above. Notice that although the NRGsolution in the left panel of figure 5.10 is half-filled, it is asymmetric. TheDMFT/DMRG calculations show a change in their behavior at approximatelythe same value of the chemical potential. In contrast to the NRG results, DMRGstabilizes a metallic antiferromagnetic state away half-filling. The filling of thesystem is 〈n〉 ≈ 0.9 and lies therefore exactly in the phase separated region inthe DMFT/NRG calculations. Although one cannot be completely sure, thereis strong evidence for phase separation between the paramagnetic metal and theantiferromagnetic insulator in this parameter regime [142–144]. The reason for


the stable antiferromagnetic metal is most likely again the interplay betweenthe broadening and deconvolution.

5.4 Frustrated Bethe Lattice

5.4.1 Half-Filling

I will now introduce frustration to the antiferromagnetic state. Frustration isone of the main arguments when explaining, why the PMIT reaches out of theantiferromagnetic dome in V2O3 (chapter 1.2.1). This leads to the situationthat not all interactions can be saturated. The simplest example is a triangle,in which all bonds are antiferromagnetic. If two of the three sites are in a Neel-state the third site has a bond to an up-site and one to a down-site. Therefore itcannot saturate both bonds. V2O3 crystallizes in a corundum structure, whichis a frustrated three-dimensional lattice [14].The calculations performed here are done for a Bethe lattice with NN- andNNN-hopping. Including both hopping terms into the Hamiltonian creates tri-angles in the lattice (see chapter 2.5). The major difference within the DMFTto NN-hopping only is that the DOS becomes now asymmetric and develops avan-Hove singularity at the lower band edge for t2/t1 > 1/4. I will first ana-lyze the results for half-filling. There were already calculations within DMFTfor a frustrated Bethe lattice. But most of the old calculation used the socalled two-sublattice frustrated model [70,147,148,165–167], which results in aparticle-hole symmetric DOS even with frustration. As side effect, this way ofintroducing frustration leaves the paramagnetic phase unchanged. For the frus-trated antiferromagnetic ordered system, on the other hand, there exists a lowercritical value for the interaction U , which increases with increasing frustration.It was furthermore found that the Neel-temperature decreases with increas-ing frustration such that the PMIT eventually outgrows the antiferromagneticphase [147]. In early calculations using this way of introducing frustrationbased on exact diagonalization studies or Hartree-Fock of the two-sublatticefully frustrated model [70, 166, 167], the authors also found parameter regionsin the phase diagram, where an antiferromagnetic metal appeared to be stable.However, this antiferromagnetic metallic phase was later traced back to numer-ical subtleties in the exact diagonalization procedure and shown to be actuallyabsent from the phase diagram [147].The first attempt to study the Hubbard model for the Bethe lattice with thecorrect inclusion of NN- and NNN-hopping has been performed rather recentlyin 2007 by M. Eckstein et al. [168] within the self-energy functional approach[169]. The authors particularly focused on the PMIT for t2/t1 = 3/7 and foundphase separation between the insulating and metallic phase.In this part of the work I will investigate the PMIT as well as the antifer-romagnetic phase at half-filling and concentrate on the competition betweenthe paramagnetic phase including the PMIT and the antiferromagnetic phaseat intermediate and high grades of frustration. I especially look at the caset2 → t1 and raise the question, if the scenario of the outgrowing PMIT, pro-posed in [147], still holds for the correct asymmetric density of states. The cal-

5.4 Frustrated Bethe Lattice 81

1 1.2 1.4U/W

0

0.002

0.004

0.006

0.008

0.01

T/W

t2/t

1=0

t2/t

1=0.6

t2/t

1=1

Figure 5.11: The transition lines for the PMIT for different frustrations as function oftemperature and interaction strength. For each frustration the right line represents thetransition from the metal to the insulator, while the left line represents the transitionfrom the insulator to the metal. Symbols mark the calculated data points, the linesare fits meant as guide to the eye.

culations were done using Wilson’s NRG as the impurity solver for the DMFT,with Λ = 2, NK = 1800 states kept per NRG step and a logarithmic broaden-ing, b = 0.8, to obtain spectral functions. Note that the paramagnetic resultsare obtained by artificially suppressing an antiferromagnetic instability. Theoccupation was kept fixed at n = 1±0.005 by adjusting the chemical potential.In contrast to the case with t2 = 0 it is not possible to achieve n = 1 here withinnumerical precision due to the asymmetric DOS. For increasing t2/t1 → 1 thePMIT is shifted towards lower interaction strengths and also lower temper-atures, see figure 5.11. While the shift in the interaction strength is rathermoderate, there is a large difference in the temperature of the critical endpointbetween the unfrustrated and highly frustrated system. A shift of the samemagnitude was also reported by M. Eckstein et al. [168]. This observation, ofcourse, renews the interest in the question, to what extent long-range hoppingcan help to push the paramagnetic MIT out of the expected antiferromagneticphase for reasonable magnitudes of t2 to create a phase diagram similar to theone found for V2O3. The scenario proposed by R. Zitzler et al. [147] relied onthe fact that the paramagnetic phase largely remains unaltered with increasingt2. As the Neel-temperature for the antiferromagnet is reduced at the sametime, the PMIT can eventually outgrow the antiferromagnetic phase. Due tothe reduction of the critical point of the PMIT this must be analyzed morecarefully now.


5.4.2 Antiferromagnetism at half-filling

I will now allow for antiferromagnetic ordering in the calculations. Figure 5.12shows the resulting phase diagrams for t2/t1 = 0.6 (upper panel) and t2/t1 =0.8 (lower panel) for different temperatures and interaction strengths. Thephase diagrams were constructed by fitting the magnetization to the actuallycalculated data points. This of course means that the phase boundaries shownhere must be considered as guess only. However, as one does not expect anystrange structures to appear, this guess will presumably represent the truephase boundary within a few percent. The full lines in figure 5.12 are thePMIT transitions. Note that for both diagrams the same division of axes waschosen.In contrast to the Hubbard model for a bipartite lattice with t2 = 0, therenow exists a finite critical value UAF

c > 0, below which no antiferromagnetismcan be stabilized even for temperature T −→ 0. With increasing frustrationthe paramagnetic-antiferromagnetic transition is shifted towards higher inter-action strengths and lower temperatures, while the PMIT is shifted towardslower interactions strengths. So obviously the PMIT is shifted towards thephase boundaries of the antiferromagnetic dome. So far this is the expectedeffect of the NNN-hopping which introduces frustration to the antiferromag-netic exchange. However, note that although t2/t1 = 0.8 represents alreadya very strongly frustrated system, the PMIT still lies well covered within theantiferromagnetic phase.I will now have a closer look at the paramagnetic-antiferromagnetic transition.Here, R. Zitzler et al. [147] made the prediction that one has to expect a firstorder transition close to the critical UAF

c at low temperatures; while at largervalues of U again a second order transition was found. Figure 5.13 shows thestaggered magnetization for different temperatures and interaction strengthsat fixed t2/t1 = 0.8. The upper panel collects data for the transition at lowtemperatures at the lower edge of the antiferromagnetic phase. The full linesrepresent the transition from the paramagnetic to the antiferromagnetic statewith increasing interaction strength for two different temperatures, while thedashed lines represent the transitions from the antiferromagnetic to the para-magnetic state with decreasing interaction strength. In the upper panel (lowT ) one can clearly see a hysteresis of the antiferromagnetic transition. Thishysteresis as well as the jump in the magnetization are clear signs for a first or-der transition. This antiferromagnetic hysteresis is very pronounced for strongfrustration but numerically not resolvable for example for t2/t1 = 0.2. Thehysteresis regions seems to shrink with decreasing t2 and eventually cannot beresolved anymore with numerical techniques. Note that such a hysteresis is alsofound in the two-sublattice fully frustrated model [147], which means that thisis quite likely a generic effect in frustrated systems at intermediate couplingstrengths. The lower panel in Fig. 5.13 shows the staggered magnetizationfor temperatures just below the corresponding Neel-temperatures and at largeinteraction strengths. Here the magnetization vanishes smoothly, which is thebehavior expected for a second order phase transition. In summary I thus finda first order transition at the critical interaction UAF

c , where the antiferro-


Figure 5.12: The upper (lower) panel shows the T − U phase diagram for t2/t1 = 0.6(t2/t1 = 0.8). The colored area represents the antiferromagnetic phase, while theblack area represents the paramagnetic phase. The lines show, where the PMIT inthe paramagnetic phase would occur. The phase diagrams were constructed by fittingthe magnetization to approximately 70 calculated data points. Additional calculationswere performed to find the PMIT-lines.


0.8 1 1.2U/W

0

0.2

0.4

0.6

0.8

1

n ↑-n↓

increasing U at T/W=1.7e-4decreasing U at T/W=1.7e-4increasing U at T/W=1.4e-3decreasing U at T/W=1.4e-3

1 1.5 2 2.5U/W

0

0.2

0.4

0.6

0.8

1

n ↑-n↓

T/W=0.006T/W=0.008

Figure 5.13: Staggered magnetization versus interaction U/W for two different tem-peratures and t2/t1 = 0.8. In the upper panel there are two transition lines for eachtemperature, representing either increasing or decreasing interaction strength. Theregion between both lines embodies a hysteresis region. The lower panel shows thetransition for large interaction strengths and high temperatures. Here no hysteresisregion can be found, but a smooth transition. The lines are meant as guide to the eye.

magnetism sets in, and a second order transition for large Coulomb parameterU ≫ W . The merging from both transition lines is an interesting point initself. There must be a critical point where the first order transition changesinto a second order transition. It is however not possible to resolve this mergingwithin DMFT/NRG. The magnetization of the system becomes very small inthis region, so it is not possible to distinguish between a (tiny) jump and nu-merical artifacts of a smoothly vanishing order parameter. Consequently, onecannot decide anymore of what order the transition is.Antiferromagnetic metallic states at half-filling were reported in earlier pub-lications [148, 165–167] and later ruled out again. In my calculations I sawno evidence for an antiferromagnetic metallic state at half-filling. Especiallyfor strong frustration t2/t1 ≈ 0.8 the system jumps directly from a paramag-netic metallic solution into an antiferromagnetic insulating solution with largemagnetization. In the papers cited, the region showing an antiferromagneticmetallic solution broadens with increasing t2. This prediction I clearly cannotconfirm, as discussed above. Only in systems with small to intermediate frustra-tion there are narrow interaction regimes where I observe a small finite weightat the Fermi level. One must however consider that the occupation number isnot exactly one but only within 0.5%, which influences the position of the gap.It was also sometimes difficult to stabilize a DMFT solution in these regions. Insummary, I cannot see any clear signs for an antiferromagnetic metallic state athalf-filling in my calculations. If any exists, then only for rather low frustrations


in a very small regime about the critical interaction UAFC . To what extent these

rather special conditions can then be considered as realistic for real materialsis yet another question.

5.4.3 Nearly fully frustrated system

In the last part concerning half-filling I want to study the situation, in whicht1 and t2 are comparable in strength. Interestingly, there has been no attemptto calculate the phase diagram on a mean-field level in the strongly frustratedmodel t2 ≈ t1. Therefore, before discussing the results of the DMFT calcula-tions for strongly frustrated systems let me try to gain some insight into thephysics I must expect, by inspecting classical spins on a Bethe lattice with NN-interaction J1 and NNN-interaction J2. In this calculation I firstly assume a

Figure 5.14: Left: Bethe lattice z = 3 with sites numbered according to the vectorspins on the right. The nearest neighbors of site 1 must lie on a circle so there is anangle θ between 1 and 2. Similarly the nearest neighbors of one of the 2-spins must lieon a circle including the 3-spins and the 1-spin.

Bethe lattice with coordination number z. The last parameter entering is theangle θ between NN-spins. Although the initial assumption that two neigh-boring spins form an angle θ may seem somewhat restrictive, I am not awareof other configurations with lower total energy [170]. I want to minimize theenergy with respect to this angle. According to figure 5.14 the NN-spins of onespin, must lie on a circle. The spins ending on the circle are all NNN-spins ofeach other. Due to the antiferromagnetic interaction J2, one can assume thatthey want to maximize the angle between them. Since there are z spins on eachcircle, they will have an angle 2π/z projected on the circle. Using now simpletrigonometry, the angle between NNN-spins γ is given by

cos γ =2 − (2R2 − 2R2 cos(2πi/z))

2= cos(θ)2 + sin(θ)2 cos(2πi/z),


0 0.2 0.4 0.6 0.8 1θ/π

0

0.5

1

1.5

2

E/J

1

J2/J

1=0.8

-0.5

0

0.5

1

1.5

2

E/J

1

J2/J

1=0.2

0 0.2 0.4 0.6 0.8 1J2/J

1

0.6

0.8

1

θ Min

/π

Energetic Minimum

Figure 5.15: Left: Energy E/J1 via the nearest neighbor angle θ for two ratios of J2/J1.Right: Nearest neighbor angle θMin, which minimizes the energy for different ratiosJ2/J1

where i runs from 0 to z − 1, giving the different positions on one circle, andR = sin(θ) is the radius of the NN-circle. Inserting this into the Hamiltonian

E = J1

∑

i,j∈NN

~Si~Sj + J2

∑

i,j∈NNN

~Si~Sj,

one finds for the energy

E/2N = J1z cos(θ)

+ J2zz−1∑

i=1

(

cos(θ)2 + sin(θ)2 cos(2πi/z))

.

Performing now the limit z → ∞ and scaling J1z → J⋆1 and J2z → J⋆

2/z, as itis done for the hopping amplitudes in DMFT, one finally obtains for the energyper lattice site

Ez=∞(θ)/(2N) = J⋆1 cos(θ) + J⋆

2 cos(θ)2.

The energy depending on θ can be seen in figure 5.15. If both interactionsJ⋆

1 and J⋆2 are antiferromagnetic the ferromagnetic configuration of the spins

represented by θ = 0 is always a maximum. The Neel-state θ = π is thestable minimum for J⋆

2 /J⋆1 < 1/2, because d2E(θ = π)/dθ2 = J⋆

1 − 2J⋆2 . The

angle minimizing the energy for J⋆2 /J

⋆1 > 1/2 is found to be θ = arccos(− J⋆

1

2J⋆2

),

corresponding to a periodically modulated spin state. In this very simplified


Figure 5.16: Ground state (T = 0) phase diagram for different strengths of frustrationas function of the interaction. The black line within the magnetic phase signalizes theend of the hysteresis region for increasing and decreasing interaction. The phase bound-aries of the incommensurate (IC) phase, white region, are only qualitative (explanationin text).

situation the ground state changes from a Neel-state for J⋆2/J

⋆1 < 1/2 to a

ground state with periodicity with longer than two sites for J⋆2/J

⋆1 > 1/2.

For the fully frustrated system J⋆1 = J⋆

2 the stable ground state configurationcorresponds to θ = 120.In DMFT calculations I can only allow for solutions commensurate with thelattice, for which I can derive a formula for the lattice Green’s function. Thishowever will possibly be inconsistent with the spin structure favored by thesystem. If, for example, I perform a calculation focusing on the ferromagneticsolution within a parameter regime, where the system wants to order antiferro-magnetically, DMFT will not converge. To investigate spin density wave stateswith periodicity of more than two lattice sites, one has to set up the correctDMFT self-consistency equations respecting the lattice structure, as can beseen in chapter 5.2.3. While for a Bethe lattice with infinite coordination num-ber and NN-hopping only it is straightforward to extend the DMFT equationsto commensurate magnetic structures with periodicity of more than two latticesites, I did not succeed in devising a scheme that allows for such calculations forsystems with NNN-hopping. The NNN-hopping makes it impossible to uniquelyidentify the connectivity of the respective sublattices. A method proposed byM. Fleck et al. [156] for the two-dimensional cubic lattice is not applicable inthis case.I thus only allowed for paramagnetic, ferromagnetic and antiferromagnetic solu-


Figure 5.17: Magnetic phase diagram T − U for t2/t1 = 0.98 for insulating startingmedium including the PMIT as blue lines. The PMIT lies within the hysteresis re-gion of the magnetic phase, but clearly outgrows it in temperature. The white regiondesignates the IC-phase.

tions in my calculations. The resulting phase diagrams for t2 → t1 are shown infigures 5.16 and 5.17. Figure 5.16 displays the ground states for different gradesof frustration and interaction strengths. For t2/t1 < 0.95 the phase diagramalways has the same structure as for small and intermediate t2. The criticalinteraction strength UAF

c necessary to stabilize the Neel-state increases and forall values above UAF

c I find an antiferromagnetic phase with a hysteresis regionat the phase boundary. For 0.95 < t2/t1 < 1 the critical value UAF

c one needsto stabilize the Neel-state increases dramatically. For t2 = t1 finally one doesnot find an antiferromagnetic Neel-state for any interaction strength U . TheDMFT calculations however indicate that in this range of t2/t1 there actuallydoes exists another magnetic phase. Namely, for sufficiently small temperaturesone obtains a finite spin polarization in every DMFT iteration. However, theDMFT does not converge to a unique state as a function of the DMFT itera-tion (see also Fig. 5.4). In the phase diagrams in figures 5.16 and 5.17 I haveleft this regime white designating an IC phase. In this parameter regime theNeel-state becomes unstable towards the behavior shown in figure 5.4. Hereone can switch between a conventional Neel-state and the IC phase by onlya small change of the interaction strength. Note, that the phase boundariesshown in the figure must be taken with some care as I cannot compare theenergies of the Neel-state and this IC phase to properly determine the phase


boundaries. As I observe precisely the same behavior for all investigated val-ues t2/t1 = 0.96, 0.97, 0.98, 0.99, 1.0 I am convinced that the ground state inthis region is an incommensurate state, as to be expected from the results forclassical spins. Similar observations also hold for finite temperatures as shownin figure 5.17, where the T − U phase diagram for fixed t2/t1 = 0.98 is dis-played. For increasing interactions and T = 0 there is first a transition froma paramagnetic metal to the IC phase and for U/W ≈ 1.6 from the IC phaseto the Neel-state. For increasing temperature the IC phase eventually becomesunstable towards the Neel-state. In Fig. 5.17 one can also see the PMIT lines.The PMIT lies within the hysteresis region of the magnetic phases but clearlyoutgrows both magnetic phases. This is the scenario described in R. Zitzler etal. [147].

5.4.4 Doped System

As t2 becomes finite the DOS becomes asymmetric and consequently the mag-netic phase diagram becomes asymmetric with respect to half-filling, too. How-ever, for sufficient small values of t2, the magnetic phase diagram will still lookvery similar to that of the unfrustrated system shown in figure 5.2, with twonotable exceptions: For the hole doped side of the phase diagram, the incom-mensurate magnetic phase sets in at smaller values of the interaction, while onthe electron doped side it starts for larger values of the interaction. Thus, forelectron doping the phase separation between the antiferromagnetic state athalf-filling and the paramagnetic state at n > 1 prevails for stronger interac-tion strengths. Already for t2/t1 = 0.2 I find no incommensurate phase on theelectron doped side for U/W < 3. As already stated above [136,147,165–167],in order to stabilize the antiferromagnetic phase for a finite NNN-hopping oneneeds a finite interaction strength UAF

c > 0.For 1/4t1 < t2 ≤ t1 one obtains a strongly asymmetric DOS showing a square-root singularity at the lower band edge. Here one can expect, and observe, aradically different phase diagram. Figures 5.18 and 5.19 show the antiferro-magnetic phase diagrams for t2/t1 = 0.6. The first picture shows the magneticphases for different chemical potentials µ/W in the same manner as in figure 5.2.In contrast to t2 = 0, in the frustrated system the antiferromagnetic Neel-statecan now be hole-doped at T = 0. Therefore, I show the same phase diagramin figure 5.19 again, in which the occupation was substituted for the chemicalpotential. The boundaries of the IC-phase must be taken with care again, asthis phase cannot be stabilized. It is interesting to note that the IC-phase isreplaced by a doped antiferromagnetic phase. On the electron doped side ofthe phase diagram, I find neither an IC-phase nor the doped antiferromagneticphase.On further increasing the frustration to t2/t1 = 0.8, see figure 5.20, the Neel-state does extend to large values of the doping (〈n〉 = 0.72), i.e. strong frustra-tion seems to stabilize the Neel-state away from half-filling. The IC phase, onthe other hand, completely vanished from the phase diagram. If one inspectsfigures 5.19 and 5.20 more closely, one sees that the antiferromagnetic stateactually sets in away from half-filling for increasing interaction strength, i.e.


Figure 5.18: Antiferromagnetic ground state (T = 0) phase diagram t2/t1 = 0.6 fordifferent chemical potentials µ and interaction strengths U . The color encodes themagnetization. The white region represents the IC phase.

Figure 5.19: The same as in figure 5.18. As the antiferromagnetic phase exists nowaway from half-filling the chemical potential was substituted by the occupation 〈n〉.


Figure 5.20: Antiferromagnetic ground state (T = 0) phase diagram for t2/t1 = 0.8.The color encodes the magnetization. The IC-phase present for t2/t1 = 0.6 has van-ished.

0.6

0.8

1

Occ

upat

ion

<n>

0 0.1 0.2 0.3 0.4chemical Potential µ/W

0

0.2

0.4

0.6

Mag

netiz

atio

n

Figure 5.21: Ground state magnetization for t2/t1 = 0.8 and U/W = 0.75. Theupper panel shows the occupation and the lower panel shows the antiferromagneticmagnetization n↑ − n↓.


Figure 5.22: Antiferromagnetic ground state (T = 0) phase diagram for t2/t1 = 1,fully frustrated system. There is an antiferromagnetic phase, which exists only awayfrom half-filling. For strong enough interaction there is a IC-phase at half-filling, whiteregion.

UAFC (n) has its minimum at n < 1. At half-filling one finds for this values of

interaction a paramagnetic metal. On the electron doped side, one only findsa paramagnetic state, which is still phase separated to the antiferromagneticstate at half-filling.As discussed previously for half-filling, for large t2/t1 > 0.96 there appearsa new phase which, motivated by a 120-order expected for a classical spinsystem at this level of frustration, can be interpreted as such a spin densitywave order. Figure 5.22 shows the phase diagram for t2 = t1, i.e. a with respectto antiferromagnetic order fully frustrated spin system. The parameter regionfor large interaction left blank denotes precisely this spin density wave state,which also can be hole doped. What is most remarkable and rather mysterious,even for the fully frustrated system I find a stable antiferromagnetic Neel-statefor fillings between 0.55 < n < 0.8. To ensure that this result is not a numericalartifact, I performed several calculations at different temperatures and withdifferent NRG parameters like discretization or states kept. However, for lowenough temperatures I always find this antiferromagnetic island. I will comeback to this point after discussing the ferromagnetic phase diagram.

5.4.5 Ferromagnetism in the Frustrated system

As already mentioned, while antiferromagnetism is the “natural” order occur-ring in single-band systems as studied here, ferromagnetic order is usually onlyobtained under more restrictive conditions. In this part I therefore want to


focus on possible ferromagnetic solutions in the frustrated system.One of the first heuristic treatments of metallic ferromagnetism was by E. Stoner[171]. He gave the criterion UDF > 1 for stabilizing ferromagnetism, where Uis the value of the on-site Coulomb interaction and DF is the value of thedensity of states at the Fermi level. Already in this criterion one sees thatferromagnetism is created by the interplay of the kinetic energy, characterizedby DF , and the Coulomb interaction, characterized by U . A rigorous resultwas obtained by Nagaoka [149], who proved the existence of ferromagnetism forU = ∞ and “one hole” for certain lattices.In the beginning of the 1990’s, A. Mielke and H. Tasaki proved the existenceof ferromagnetism under certain conditions on the dispersion, known as “flatband ferromagnetism” [172,173]. Here the ferromagnetic ground state appearsdue to a dispersion-less (flat) lowest lying band. This flat band introduces ahuge degeneracy of the ground state for U = 0, which is lifted by the Coulombinteraction. A nice overview about this topic and other rigorous results forferromagnetism can be found in the work of H. Tasaki [174].Remembering the singularity in the DOS for t2/t1 > 0.25, the situation presentin this system is very similar to the “flat band” scenario. Former studies foran asymmetric DOS [175–178] already showed the existence of ferromagnetismin such situations. Consequently, one has to expect ferromagnetism in this sys-tem, too. Indeed, Figure 5.23 shows the ferromagnetic polarization p =

n↑−n↓

n↑+n↓,

color encoded over the occupation n = n↑ + n↓ and the interaction strengthat low temperature (T/W = 2 · 10−4). The NNN-hopping for this system ist2/t1 = 0.6. The ferromagnetic state is fully polarized. One sees, that thesingularity in the DOS alone cannot create ferromagnetism. Here one againneeds a finite interaction strength of approximately U/W ≈ 0.3, which howeveris a realistic number for transition metal compounds of both the 3d and 4dseries. In figure 5.24 I depict the lower and upper critical occupation betweenwhich the ferromagnetic state is stable as function of the interaction strength.Below the lower critical occupation, the DMFT simulations do not convergeindependent of the interaction strength. I believe that this is a numerical prob-lem due to the singularity in the DOS: If the Fermi level lies very close to thesingularity, which is the case for n→ 0, the slope of the DOS at the Fermi levelis very large. Small differences in the position of structures in the interactingGreen’s function will consequently have a great influence. However, I cannotrule out the possibility of the existence of another phase in this regime. Theoccupation number jumps in this region between almost zero and a larger value,and cannot be stabilized. The behavior can only be seen at low temperaturesand for t2/t1 > 0.25, where the singularity in the DOS is sufficiently strong andnot smeared out by temperature broadening.At the upper critical occupation and low interaction strengths the system jumpsfrom a fully polarized ferromagnet to a paramagnetic phase. For strong interac-tion the upper occupation is large enough such that the system directly changesfrom a ferromagnetic state into the incommensurate phase or the Neel-phase.As I already noted, the “flat band” scenario indicates that the ferromagneticstate is intimately connected to the appearance of the van-Hove singularity at


Figure 5.23: Ferromagnetic ground states (T ≈ 0) for t2/t1 = 0.6. Color coded is thepolarization p = (n↑ − n↓)/(n↑ + n↓).

0 1 2 3Interaction U/W

0

0.2

0.4

0.6

0.8

1

Occ

upat

ion

<n>

lower nc

upper nc

Figure 5.24: Upper and lower critical occupation between which the ferromagneticphase exists for t2/t1 = 0.6 and T = 0.


0 0.1 0.2 0.3 0.40.5Occupation <n>

0

0.2

0.4

0.6

0.8

1

Pol

ariz

atio

n U/W=1U/W=3

0

0.2

0.4

0.6

0.8

1

Pol

ariz

atio

nU/W=4U/W=6

0 0.1 0.2 0.3 0.40.5 0.6Occupation <n>

U/W=1U/W=3

U/W=1U/W=3

t2/t

1=0.21 t

2/t

1=0.24

t2/t

1=0.26 t

2/t

1=0.3

Figure 5.25: Ferromagnetic polarization for T/W = 2 ·10−4 for t2/t1 as the singularitymoves into the band. The upper panels show plots for a DOS without singularity.Note that with increasing t2/t1 the interaction one needs to stabilize the ferromagnetdecreases.

the lower band edge. I will therefore look somewhat closer on the relation ofthe formation of a ferromagnetic state and the appearance of the singularity inthe DOS. Figure 5.25 shows the polarization versus the occupation for differentNNN-hopping t2/t1 and interaction strengths close to the appearance of thesingularity in the DOS. The upper panels represent a situation where thereis no singularity present in the DOS. The interaction needed to stabilize aferromagnetic state in these systems without singularity is strongly increased.For the case of t2/t1 = 0.2 I found no ferromagnetic phase for interactions asstrong as U/W ≈ 10. As soon as t2/t1 > 0.25, the critical interaction strengthlies below U/W = 1. Increasing NNN-hopping t2 as well as increasing theinteraction strength favors the ferromagnetic state as it becomes more extendedto larger occupations. In the DMFT/QMC study of J. Wahle et al. [175] apeak at the lower band edge was enough to stabilize a ferromagnetic phaseat moderate interaction strengths. In my calculations the tendency towardsferromagnetism dramatically decreases for a DOS without singularity.

5.4.6 Energies for the different magnetic states

A careful look at the phase diagrams reveals, that there are parameter regions,where one seemingly can obtain both an antiferromagnetic as well as a fer-romagnetic solution to the DMFT equations. This is rather unusual becauseconventionally DMFT will show oscillating behavior if one performs a ferro-magnetic calculation in a regime with antiferromagnetic ground state and vice


versa.To decide which of the two solutions is the thermodynamical stable state, onehas to compare their respective free energies. As the calculations were donepractically at T = 0, I calculate the energy of the system, given by

〈H〉N

=〈HT 〉N

+U

N

∑

i

〈ni↑ni↓〉,

where HT is the kinetic energy and N the number of sites. The interactionterm U is purely local and thus can be taken from the converged impuritycalculation.The kinetic energy, on the other hand, can be calculated from the expression

〈HT 〉N

=

∞∫

−∞

dθǫ(θ)ρ(θ)

0∫

−∞

dω

(

− 1

π

)

Im1

ω + µ− ǫ(θ) − Σ(ω + iη, θ),

where Σ(ζ, θ) is the lattice self-energy, θ a suitable variable to label the single-particle energies on the lattice under consideration, and µ the chemical po-tential. Within DMFT, the lattice self-energy is approximated by a local self-energy, i.e. one may set Σ(ζ, θ) = Σ(ζ). Furthermore, for the Bethe latticewith infinite coordination ǫ(θ) = t1θ + t2(θ

2 − 1) and ρ(θ) = 12π

√4 − θ2 holds.

Substituting ǫ(θ) by ǫ in the integral, the resulting DOS takes on the form givenin chapter 2.5.Since the Neel-state is defined on an AB lattice, one has to distinguish betweenthe inter- and intra-sublattice hopping terms, and the formula for the kineticenergy takes on the form

〈HT 〉N

= − 1

πIm

∞∫

−∞

dθρ(θ)

0∫

−∞

dω(

t1θ (GAB(ω + iη) +GBA(ω + iη))

+t2(θ2 − 1) (GAA(ω + iη) +GBB(ω + iη))

)

.

Note that with the definition of the matrix Green’s function this formula canbe put into the compact matrix form

〈HT 〉N

= − 1

πIm

∞∫

−∞

dθǫ(θ)ρ(θ)

0∫

−∞

dω∑

ij

[

G(ω + iη)]

ij

[

G(ω + iη)]

ij:=

(

ζ↑ − t2(θ2 − 1) −t1θ

−t1θ ζ↓ − t2(θ2 − 1)

)−1

ij

ζσ(ω) := ω + µ− Σσ(ω + iη)

The energies of the converged solutions for t2 = t1 and U/W = 2.5 can beseen in figure 5.26. The antiferromagnetic solution could be stabilized in thisparameter region for occupations 0.55 < n < 0.8. From figure 5.26 it becomesnow clear that the ferromagnetic state has the lowest energy for n < 0.6. For0.6 < n < 0.75 the antiferromagnetic state takes over as the ground state, but

5.5 Summary 97

0.5 0.6 0.7 0.8Occupation <n>

-0.09

-0.085

-0.08

Ene

rgy

<H

>/(

NW

)

ParamagnetFerromagnetAntiferromagnet

Figure 5.26: Ground state energies for paramagnetic, antiferromagnetic, and ferromag-netic states in the fully frustrated system for U/W = 2.5. The energy was divided bythe number of sites and the non-interacting bandwidth W .

is nearly degenerate with the paramagnetic state. For fillings larger than 0.8no staggered magnetization can be stabilized any more.Thus, the energy calculations reveal two things. Firstly, an antiferromagneticNeel-state indeed seems to form away from half-filling in the fully frustratedsystem. Secondly, the energy differences are extremely small, in particular theantiferromagnet and paramagnet are de facto degenerate over the full parameterregime where the former exists.To understand this at first rather irritating observation let me recall the well-known fact that in strongly frustrated systems it is a common feature to have alarge number of degenerate ground state configurations, which also can includemagnetically ordered ones [174]. Thus, the degeneracy of the antiferromagnetand the paramagnet hints towards the possibility that there may exist a largernumber of other magnetically ordered states in this parameter region. Unfor-tunately I am not able to search for and in particular stabilize those magneticphases with the technique at hand.

5.5 Summary

In this chapter I analyzed the magnetic properties of an one-orbital Hubbardmodel within the DMFT for a Bethe lattice with infinite coordination. Theunfrustrated Bethe lattice with semi-elliptic DOS shows antiferromagnetic be-havior at T = 0 only exactly at half-filling. Interestingly, the doping dependenceof the antiferromagnetic phase changes from phase separation between the an-tiferromagnetic insulator at half-filling and the paramagnetic metal away fromhalf-filling for weak interactions towards an inbetween lying incommensurate


spin-density phase for strong interactions. This change happens exactly at thesame interaction strength at which in the paramagnetic phase the metal insu-lator transition occurs. This means that one can see signs of the PMIT even inthe antiferromagnetic phase, which actually completely covers the PMIT. Onemust note, however, that for finite T this transition does not trace the PMIT,which maybe due to a stabilization of a doped antiferromagnetic phase, butcreates room for discussion.Furthermore, I proposed a way how to stabilize such spin-density waves for aBethe lattice. Unfortunately, one has to test for each period separately and thecalculation seem to be rather unstable.For the frustrated Bethe lattice the magnetic phases change. The phase diagrambecomes asymmetric towards half-filling. While for hole-doping an antiferro-magnetic phase exists away from half-filling, for electron doping the phase sep-arated region becomes more and more extended. For increasing frustration theantiferromagnetic doped state replaces finally the incommensurate spin densitywave at the hole doped side of the phase diagram. Additionally, one now needsa finite interaction strength UAF

C > 0 for stabilizing the antiferromagnetic state.Interestingly, for the fully frustrated system, there still exists an antiferromag-netic Neel-phase away from half-filling. The Neel-state can be stabilized in thesame parameter region as the ferromagnetic state. Comparing the energies ofthe possible states, there seems to be a transition from the ferromagnetic to theantiferromagnetic state. But the latter one is energetically almost degenerateto the paramagnetic state. Thus frustration seems to stabilize different statesin this parameter region. It is an intriguing possibility, that there could bemore degenerate magnetic states in this regime, stabilized due to the frustra-tion. As I am, however, only able to look for homogeneous or Neel-states, thisis only speculative, nevertheless motivating further studies of magnetic orderin the single-band Hubbard model with different methods to solve the DMFTequations. However, for these studies the Bethe lattice may not be a suitablechoice any more, as the definition of a wave vector ~Q to identify the variouspossible spin structures is not possible here.

CHAPTER 6

Two-Orbital Hubbard Model

6.1 Introduction

Finally, I will present results for the two-orbital Hubbard model, describing twocorrelated degenerate bands. It was introduced in chapter 1.3, reading

H = HT +HU

HT =∑

ij

2∑

m=1

∑

σ

tij,mc†i,m,σcj,m,σ,

HU = U∑

i

2∑

m=1

ni,m,↑ni,m,↓ +

(

U ′ − 1

2J

)

∑

i

2∑

l,m=1

l<m

∑

σ,σ′

ni,l,σni,m,σ′

−2J∑

i

∑

l<m

~Sil · ~Sim.

The operator HT represents the kinetic energy corresponding to the hopping ofthe electrons with amplitude tij. I assume throughout this chapter that there isonly nearest neighbor hopping and that the hopping amplitudes are degeneratefor both bands resulting in equal bandwidths W = 4t. The calculations wereagain performed for a Bethe lattice with semi-elliptic density of states. Theinteraction HU consists of three different terms: an intra-band density-densityinteraction with amplitude U , an inter-band density-density interaction withamplitude U ′− J

2 , and a spin-spin interaction with amplitude −2J , representingthe ferromagnetic Hund’s coupling for J > 0. In the calculations shown inthis chapter, I mainly concentrate on U/W = 4, which is a good guess fortransition metal oxides [9]. I will vary the Hund’s coupling J and the inter-orbital interaction U ′ and analyze their effects for different occupations of thesystem.

100 Two-Orbital Hubbard Model

The two-orbital Hubbard model is an appropriate model for describing theelectronic correlations in manganites, as introduced in chapter 1.2.4. The phasediagram of La1−xCaxMnO3 can be seen in figure 1.6. It shows a very rich phasediagram consisting of magnetic and orbitally ordered phases. Due to the cubiccrystal symmetry the d-orbitals split into a threefold degenerate t2g- and atwofold degenerate eg-band, which for the coordination present in perovskiteshas the higher energy compared to the t2g-band. In La1−xCaxMnO3 one hasto distribute 4 − x electrons per site to the d-states according to Hund’s rules.Thus, the electronic situation in this compound can be modeled by a partiallyfilled eg-band below quarter-filling and a half-filled t2g-band, which couples viaHund’s coupling ferromagnetically to the eg-electrons. The hopping between thet2g-states is very small and thus the t2g-states are often modeled as localizedS = 3/2 spins. Besides this electronic part the lattice degrees of freedom andespecially Jahn-Teller distortions are important to correctly describe the physicsof manganites.The aim of this chapter is to understand the role of the electronic degreesof freedom in the complex phase diagram of the manganites. In particular, towhat extent the strongly correlated eg-band is sufficient to reproduce the generalfeatures. Thus the investigations done here completely neglect the t2g-spin andthe coupling to the lattice. Based on such an investigation, one can includeadditional interactions, like the coupling to the t2g-spin (or band) respectivelythe strong interaction with the lattice step by step, properly identifying forwhich particular features they are actually responsible. As I will discuss, alreadythis simplified situation shows very complex ground state properties, involvingthe coupling of charge, spin and orbital degrees of freedom [179].In the second section of this chapter, I will show calculations for two site clusters,in which the whole lattice consists of only two sites. The third section describesthe magnetic ground state diagram of the two-orbital Hubbard model. In thissection I focus on occupancies larger than one electron per lattice site. Thefinal section concentrates on the quarter-filled system. Quarter-filling is a veryspecial situation for the two-orbital Hubbard model. Exactly at quarter-filling,I can observe four different long range ordered phases. There is a competitionof two different ferromagnetic states, an antiferromagnetic state, and a chargeordered state. Between the two ferromagnetic states one can find a metalinsulator transition (MIT). These results are submitted for publication [180].

6.2 Two Site Cluster

Before studying the Hubbard model for a Bethe lattice within the DMFT, Iwill present some calculations for a two site cluster. This means that the wholelattice consists of only two sites, each built up from two orbitals. Altogetherthe system can be occupied by 8 electrons. But as the system is particle holesymmetric, I will only discuss the subsystems with 4 or less electrons. Eventhough one cannot expect to see the whole physics of the Hubbard model, onecan get some insight into what one can expect.The ground state of four electrons for a two site cluster, meaning half-filling, is

6.2 Two Site Cluster 101

0 0.25 0.5 0.75 1Hund’s coupling J/W

-0.5

-0.4

-0.3

Ene

rgy

Sz=1/2Sz=3/2

Figure 6.1: Exact diagonalization of a two site cluster with t = −0.05, U = 0.8,U ′ = 0.6, µ = −0.3 and 3 electrons in the system. For small Hund’s coupling the groundstate is a low spin configuration, while for large Hund’s coupling the low and highspin configurations are degenerate. For scaling the x-axis, I have used the bandwidthW = 0.2 of the free DOS in the next section.

0 0.4 0.8 1.2 1.6Hund’s coupling J/W

-0.63

-0.62

-0.61

-0.6

Ene

rgy

Sz=0, Tz=0Sz=0, Tz=1Sz=1, Tz=0Sz=1, Tz=1

Figure 6.2: Exact diagonalization of a two site cluster with t = −0.05, U = 0.8,U ′ = 0.6, µ = −0.3 and 2 electrons in the system. The states with one electron in eachorbital are degenerate ground states. I used again W=0.2 for scaling the x-axis.


an antiferromagnetic state due to super exchange. Of course, one must be care-ful with the expression “antiferromagnetic” in a two site cluster. The groundstate is a superposition of several possible states. The principal contributionscome from the states, where one site consists of two up-electrons and the otherof two down-electrons, and its interchanged configuration. If for a real lattice,like the Bethe lattice, long range order can persist, critically depends on thelattice and the temperature. For example, as seen in chapter 5, frustration canprevent the system from forming an antiferromagnetic Neel-state. But in thecalculations for a two site cluster, the states, from which an antiferromagneticNeel-state would be built up, possess the lowest energy.Figure 6.1 shows the ground state energies of the two site cluster with threeelectrons in the system. The interaction parameters are similar to the param-eters, which are chosen in the next sections. The Hund’s coupling is varied tosee its influence on the ground state energy. The configuration with all threeelectrons in the same orbital has the highest energy, as one state must be dou-ble occupied and the local intra-band interaction U is the largest interactionparameter. The remaining two configurations are such that two electrons arein the same orbital and the third electron occupies the other orbital. There isone ferromagnetic state with all spins aligned and therefore Sz = 3/2, and onemixed state with Sz = 1/2. Of course, the configurations with interchangedSz-components, Sz = −1/2 and Sz = −3/2, exhibit the same energy. Thereis a level crossing in the quantum number subspace of Sz = ±1/2 at approxi-mately J/W = 0.5. For Hund’s coupling J below this value, the ground stateis a doublet of Sz = ±1/2 states. Above this value the ground state is fourfolddegenerate consisting of one state from the Sz = ±1/2 subspace and one statefrom the Sz = ±3/2 subspace each. The states from the Sz = ±3/2 subspacerepresent the possibility of a ferromagnetic state in an extended system. Thisshows the importance of the Hund’s coupling J for ferromagnetism in this sys-tem. One will see in the next section, that also in an extended system there is aground state transition towards a ferromagnetic phase for occupation 〈n〉 ≈ 1.5when increasing J .In figure 6.2 one can see the ground state energies for the case, when there arealtogether two electrons on both sites, representing quarter-filling. There arefour quantum number subspaces under consideration. Quantum number Sz = 0corresponds to the situation of one up electron and one down-electron, whilefor Sz = 1 both electrons possess spin component Sz = 1/2. The subspacewith both electrons having Sz = −1/2 is degenerate to the last one. The samediscrimination can be done for the orbitals. As the pair hopping term wasneglected in this work, one can introduce a conserved orbital quantum numberTz. Tz = 0 means, that one electron is in the orbital, represented by m = 1in the Hamiltonian, and the other electron occupying m = 2. For Tz = 1both electrons occupy the same orbital. The ground state energies in figure 6.2show that there is a degenerate ground state with Tz = 0. The configurationcorresponds to a triplet state in the spin space, but the Sz = −1 state is notshown here. In an extended bipartite lattice this configuration can result in aNeel-state for the orbital occupation: On the A-sites of the lattice the electronssit on the m = 1 orbital, while on the B-sites the electrons sit on m = 2. I will

6.3 Magnetic Phase Diagram 103

0 0.01 0.02 0.03Temperature T/W

0

0.5

1

1.5

2

Mag

netiz

atio

n

J=W

Figure 6.3: Antiferromagnetic magnetization m = n↑ − n↓ versus the temperature forU/W = 4, J/W = 1, and U ′/W = 2 at half-filling. The transition does only change alittle upon varying U ′ or J .

show results on such states later in this chapter about quarter-filling.Calculations with altogether only one electron are not shown here, as all inter-action terms vanish. Thus all states possess the same energy of E = −µ+ t.

6.3 Magnetic Phase Diagram

For an extended system like the Hubbard model for a Bethe lattice the situationis different from the two site cluster. Even though the ground state of the twosite cluster consists of aligned spins, it is a priori not clear, if the ground stateis ferromagnetical for an infinite lattice. I will here only present results forthe semi-elliptic free density of states (DOS) of the infinite dimensional Bethelattice, corresponding to a bipartite situation. All results are obtained withinthe DMFT with the NRG as impurity solver. The two-orbital model is anextreme case for using the NRG. The calculations were performed parallelizedon eight-core multiprocessor architectures using up to 15 GB memory. For allcalculation the NRG parameters Λ = 2.0 and NK = 4000 − 5000 were used.As the intra-orbital density-density interaction U is usually the strongest in-teraction present, the system is a good candidate for super exchange at half-filling [51] (chapter 1.4). And indeed, the system forms an antiferromagneticinsulator at half-filling. The temperature dependence of the antiferromagneticmagnetization m = n↑ − n↓ is shown in figure 6.3. With increasing tem-perature the magnetization vanishes smoothly, signaling a second order phasetransition. The transition temperature and the characteristics of the transi-tion mainly depend on the intra-orbital interaction U . As in the one-orbital


Figure 6.4: Magnetic ground state phase diagram, U/W = 4, T/W = 2 · 10−4, for〈n〉 = 1.5 (upper panel) and 〈n〉 = 1.2 (lower panel). For stabilizing a ferromagneticphase, one needs a finite Hund’s coupling J > 0. For 〈n〉 = 1.5 the incommensuratephase extending from half-filling, is still present. This phase has vanished for 〈n〉 =1.2. The points denote the parameters, at which calculations were done. The phaseboundaries are only schematic.

6.3 Magnetic Phase Diagram 105

Hubbard model for the unfrustrated Bethe lattice, this antiferromagnetic Neel-state cannot be doped. The antiferromagnetic state becomes unstable towardsan incommensurate spin density wave, when changing the chemical potentialµ. This was already shown for such strong interaction strengths in chapter 5for the one-orbital Hubbard model.The phase diagram of the one-orbital Hubbard model with semi-elliptic DOS,corresponding to a Bethe lattice without frustration, does not show any otherordered phases. In contrast, there are parameter regions in the two-orbital Hub-bard model, which are dominated by double exchange (chapter 1.4), showinglarge regions of ferromagnetism [33, 181–187]. I have shown in the last sectionfor three electrons in a two site cluster, that there is a ground state transitionwhen increasing J and that the state with parallel aligned electrons becomesthe ground state for large enough ferromagnetic Hund’s coupling J . The sameholds true for the Bethe lattice. For 〈n〉 = 1.5, there is a transition from theincommensurate phase to a ferromagnetic phase for increasing Hund’s coupling,see figure 6.4. The incommensurate phase shows the same non-convergent be-havior in the DMFT as described in the last chapter. This phase extends upto half-filling, where it is displaced by the antiferromagnetic Neel-state. For〈n〉 = 1.2 the incommensurate phase has completely vanished. This phase ismost probably due to antiferromagnetic tendencies at half-filling, which arein the doped system incommensurable with the filling of the lattice and van-ish when doping too far away from half-filling. The ferromagnetic state, on theother hand, is stabilized by the Hund’s coupling J . Interestingly, strong Hund’scoupling alone is not enough. For U/W = 2 the ferromagnetic tendency in theparameter region for 〈n〉 = 1.2 − 1.5 decreases rapidly. For U/W = 2 it is onlypossible to stabilize a ferromagnetic phase for J/W = U ′/W = 2, which, how-ever, is a questionable large Hund’s coupling J compared to U . For lower valuesof the Hund’s coupling and U/W = 2 the phase diagram for 〈n〉 = 1.0 − 1.5is dominated by a paramagnetic metallic phase. In contrast, for U/W = 4the ferromagnetic phase covers most of the magnetic phase diagram for fill-ing 〈n〉 = 1.2. Only for low Hund’s coupling the paramagnetic phase is theonly stable one. The ferromagnetic phase diagrams for the whole occupationand temperature range and U/W = 4, J/W = 1, U ′ = 2 (upper panel) andU/W = 4, J/W = 1.5, U ′ = 1 (lower panel) can be seen in figure 6.5. In bothdiagrams the ferromagnetic polarization p =

n↑−n↓

n↑+n↓is color coded. The white

region denotes the already mentioned incommensurate phase. For better com-parison, the same division of axis was chosen in both diagrams. Therefore, onecan see how the ferromagnetic phase becomes more extended to larger valuesof hole doping when increasing the Hund’s coupling. For J/W ≥ 1.5 the fer-romagnetic phase extends to fillings less than quarter-filling 〈n〉 = 1. This willbecome important in the next section. So far, the magnetic phase diagram ofthe two-orbital Hubbard model consists of an antiferromagnetic phase at half-filling, which becomes unstable towards an incommensurate spin density wavewhen doping, and eventually changes to a ferromagnetic state at occupation〈n〉 ≈ 1.5 if the Hund’s coupling J is large enough.


Figure 6.5: Magnetic ground state phase diagram for U/W = 4, J/W = 1, U ′/W = 2(upper panel) and U/W = 4, J/W = 1.5, U ′/W = 1 (lower panel). The ferromagneticpolarization p =

n↑−n↓

n↑+n↓is color coded. The white region corresponds to the incommen-

surate phase. In this parameter region the DMFT calculations did not converge. Bothphase diagrams were created by fitting of approximately 30 points each. Notice thatfor both diagrams the same division of axis was chosen.

6.4 Quarter-Filling 107

6.4 Quarter-Filling

6.4.1 Ferromagnetic Metal Insulator Transition

Quarter-filling, like half-filling, represents a very special point for the two-orbitalHubbard model. In a classical picture there is one electron per site, whichcan choose between two orbitals. This picture already makes clear that orbitaldegeneracy and fluctuations play an important role for quarter-filling. This wasconfirmed by the two site cluster calculation of the last section. In figure 6.6,I show the ferromagnetic ground state phase diagram for 〈n〉 = 1 as functionof U ′ and J . The intra-orbital interaction was U/W = 4. As noted before, forstrong Hund’s coupling one obtains the orbitally homogeneous ferromagneticphase discussed in figure 6.5. However, for large enough repulsive inter-orbitaldensity-density interaction U ′, I observe that the ferromagnetic spin alignmentis accompanied by an antiferro-orbital order of the conventional Neel-type [182,183,187–191]. The phase diagram at 〈n〉 = 1 is mainly covered by ferromagneticphases. Only for small values of the Hund’s coupling a paramagnetic phase canbe found as a ground state. Figure 6.7 shows the spectral functions of both

Figure 6.6: Ferromagnetic phase diagram for U/W = 4 and 〈n〉 = 1. The calcula-tions were done at the parameters denoted by the black points. Therefore the phaseboundaries are only schematic.

ferromagnetic states. The left (right) panel shows the configuration and thespectral function of the homogeneous (orbitally ordered) phase. These resultswere obtained at T/W = 2·10−4. The Fermi energy corresponds to ω = 0. Thusone can observe a dramatic difference in the behavior of both states at the Fermienergy. The majority spin of the homogeneous ferromagnetic state has largespectral weight at the Fermi level, showing metallic behavior. In contrast, theorbitally ordered ferromagnetic state possesses a gap at the Fermi energy. Thereis no spectral weight. Thus the orbitally ordered ferromagnet is an insulator.


Figure 6.7: Spectral functions and schematic occupation of the orbitals within bothferromagnetic phases. Left: Homogeneous ferromagnetic state at J/W = 1.5 andU ′/W = 1. Right: Orbitally ordered ferromagnetic state at J/W = 0.5 and U ′/W = 3.Note the different axis divisions.

The evolution of the spectral function of the orbitally ordered ferromagneticstate and the gap size can be seen in figure 6.8. One can observe how thegap size linearly shrinks and finally closes at U ′/W ≈ 1.4. Thus, by varyingHund’s coupling J or U ′ one can observe a metal insulator transition (MIT)between two almost fully polarized ferromagnetic phases. Note that this metalinsulator transition is very different from the usual paramagnetic one, whichappears in the Hubbard model as function of U in the paramagnetic state. Theobserved MIT within the ferromagnetic phases, however, is due to a strong inter-orbital density-density interaction, which is responsible for driving the orbitalordering. The usual intra-orbital Hubbard interaction U plays a minor role inthis transition. There is a first order transition between the orbitally orderedferromagnetic state and the homogeneous one: both the magnetic and orbitalpolarization show jumps when crossing the phase boundary, which can be seenin figure 6.9. Note that both states are nearly fully polarized; the orbitallyordered ferromagnetic state has p =

n↑−n↓

n↑+n↓= 1, while the homogeneous has

p = 0.82. This means that the jump in the magnetization is comparativelysmall. A more important aspect is that the orbitally ordered ferromagneticphase is an insulator, while the homogeneous one is a metal.Both states show very different dependence on the chemical potential. Forthe homogeneous ferromagnetic phase I already showed, when discussing figure6.5, that its filling can be varied smoothly. The dependence on the chemicalpotential of the orbitally ordered ferromagnet on the other hand is completely


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Frequency ω/W

0

1

2

3

4

5

ρ(ω

)W

U’/W=2.25U’/W=2U’/W=1.75U’/W=1.5U’/W=1.45

1 1.5 2 2.5U’/W

0

0.2

0.4

0.6

0.8

gap

size

/W

linear fit

Figure 6.8: Top: Evolution of the spectral function of the orbitally ordered ferromag-netic state for U/W = 4 and J/W = 1. Bottom: Size of the gap extracted from thespectral functions as function of U ′. The red line corresponds to a linear fit.

1 1.2 1.4 1.6 1.8 2U’/W

0

0.2

0.4

0.6

0.8

1

Occ

upat

ion

Polarization nup

-ndown

nup

upper orbital

nup

lower orbital

Figure 6.9: Dependencies of the occupations on U ′ near the metal insulator transitionfor U/W = 4 and J/W = 1. The black points denote the polarization of the ferromag-netic state. The red and green points show the occupation for the majority spin forthe upper and lower orbital.


0.8

1

Occ

upat

ion

0 0.5 1 1.5 2 2.5chemical potential µ/W

00.20.40.60.8

1

Pol

ariz

atio

n

inco

mm

ensu

rate

Figure 6.10: Dependence of the orbitally ordered ferromagnetic state for J/W = 0.5and U ′/W = 3 upon changing the chemical potential µ. The upper (lower) panel showsthe occupation (polarization) of the system.

different and can be seen in figure 6.10. The plot shows that for a criticalchemical potential the filling of the system jumps from approximately 〈n〉 = 0.88to quarter-filling 〈n〉 = 1. At the same critical chemical potential also thepolarization jumps from a non-polarized state to a nearly fully polarized. Inother words, there is a first order transition between a paramagnetic phase withfilling less than one and an orbitally ordered ferromagnetic state at quarter-filling. Consequently the electronic system shows phase separation betweenthese states and the precise physics will depend on additional interactions,like long-range Coulomb interaction, or additional degrees of freedom like thelattice. For electron doping this orbitally ordered ferromagnet becomes unstabletowards an incommensurate spin density wave. Unfortunately, it is very hardto stabilize such a spin density wave in our calculations, so that I am not ableto study the exact nature and energetic stability of this phase.With respect to effects like the colossal magneto resistance effect in manganites,introduced in chapter 1.2.4, it should be very interesting to look at the temper-ature dependence of the orbitally ordered ferromagnetic state. In manganites aferromagnetic metal becomes unstable towards a paramagnetic insulating statewhen increasing the temperature. The transition temperature can be tunedby applying a magnetic field, which explains the large resistance change whenvarying the magnetic field. The metal insulator transition observed in the two-orbital Hubbard model provides the opportunity to see similar effects. Thequestion is, whether this MIT can be triggered by applying a magnetic field orchanging the temperature. Figure 6.11 shows the temperature dependence of astate deep within the orbitally ordered ferromagnetic phase. The ground state of


-4 -2 0 2 4ω/W

0

0.5

1

1.5

2

ρ(ω

)W

-2 0 2 4ω/W

-2 0 2 4ω/W

-2 0 2 4ω/W

upper orb majority spinupper orb minority spinlower orb minority spinlower orb majority spin

T/W=2e-4 T/W=3.6e-3 T/W=1.5e-2 T/W=3e-2

Figure 6.11: Temperature dependence of the orbitally ordered ferromagnetic state forJ/W = 0.5 and U ′/W = 3 for four different temperatures. For T/W = 1.5 · 10−2,both spin directions for each orbital are degenerate, denoting an orbitally orderedparamagnet. For T/W = 3 · 10−2, all four lines lie above each other.

-1 -0.5 0 0.5 1ω/W

0

0.5

1

1.5

2

ρ(ω

)W

-0.5 0 0.5 1ω/W

-0.5 0 0.5 1ω/W

lower orbital minority spinupper orbital minority spinlower orbital majority spinupper orbital majority spin

T/W=3.6e-3 T/W=1.5e-2 T/W=3e-2

Figure 6.12: Temperature dependence of the orbitally ordered ferromagnetic state forJ/W = 1 and U ′/W = 1.45 for three different temperatures. In the second panel, bothorbitals are degenerate. In third panel, all lines lie above each other.


Figure 6.13: Proposed schematic T -U ′ phase diagram for quarter-filling. The followingabbreviations are used: PM = paramagnetic meta; PI = paramagnetic insulator; OPI= orbitally ordered paramagnetic insulator; OFI = orbitally ordered ferromagneticinsulator; FM = ferromagnetic metal.

the system is the fully polarized orbitally ordered ferromagnetic state, alreadyshown above. When increasing the temperature, the ferromagnetic polariza-tion vanishes, but the orbital order remains up to T/W ≈ 2 · 10−2. Finally theorbital order becomes unstable towards a paramagnetic insulator. Obviously,the orbital order is the stronger ordering mechanism, persisting longer thanthe ferromagnetic order when increasing the temperature. But as the systemremains in an insulating state, no temperature triggered ferromagnetic MITcan be observed in this parameter region. The insulating paramagnetic regionreaches as high as T/W ≈ 0.4. For a system with bandwidth W = 0.5 eV,this would imply a transition temperature of about T ≈ 2400K. Figure 6.12shows the spectral functions in the orbitally ordered ferromagnetic phase nearthe ferromagnetic metal insulator transition for different temperatures. Theorbital ordering now vanishes before the ferromagnetic transition occurs, and aferromagnetic metallic state is stabilized. Finally, this state becomes unstabletowards a paramagnetic metallic state. When increasing the temperature fora state in the homogeneous ferromagnetic phase, the ferromagnetic polariza-tion vanishes and a paramagnetic metal is formed. Therefore, the transitiontemperatures of the ferromagnetic order and the orbital order change quiteindependently when varying the interaction parameters. Additionally, therehas to be a paramagnetic MIT for high temperatures. Figure 6.13 shows aschematic phase diagram, which can be drawn after the just discussed results.Of course, the characteristics of the transition lines are here just proposed, butthey include both temperature evolutions shown in the figures 6.11 and 6.12.For high values of the inter-orbital density interaction U ′, there is the para-magnetic metal insulator transition. For lower temperatures a transition tothe orbitally ordered paramagnet and the orbitally ordered ferromagnet can befound. In contrast, for inter-orbital density interaction U ′ close to the vanishingof the orbital order at T = 0, one can see the temperature driven transition


0.9

0.95

1

1.05

1.1

Occ

upat

ion

0 0.5 1 1.5 2 2.5chemical potential µ/W

0

0.2

0.4

0.6

0.8

1

Pol

ariz

atio

n

-4 -3 -2 -1 0 1 2 3 4ω/W

0

0.2

0.4

0.6

0.8

1

ρ(ω

)W

Minority spinMajority spin

Figure 6.14: Antiferromagnetism for J/W = 0.5 and U ′/W = 3. The left panel showsthe doping dependence of the occupation and polarization for the antiferromagnet state.The right panel shows the spectral function.

from the paramagnetic metal at high temperatures to the ferromagnetic metaland later eventually to the orbitally ordered ferromagnetic insulator.

6.4.2 Antiferromagnetism and Charge Order

In the region of large U ′, where the orbitally ordered ferromagnet is found, itwas also possible to stabilize an antiferromagnetic Neel-phase at quarter-filling.As the system is dominated by ferromagnetic double exchange at quarter-filling,one actually does not expect such a phase here. An antiferromagnetic phaseat quarter-filling was reported by T. Momoi [183] using an antiferromagneticHund’s coupling. Like the orbitally ordered ferromagnet, this antiferromagneticphase also exists only exactly at quarter-filling. The spectral function and thedoping dependence can be seen in figure 6.14. The spectral function shows thatalso this state is a perfect insulator. When trying to dope it, one again findsphase separation to a paramagnetic metal away from quarter-filling. In orderto find out which state is the thermodynamically stable one, I calculated theenergy of both states. The result is that, as expected, the orbitally orderedferromagnet has the lower energy, thus is the thermodynamically stable one.Nevertheless, the calculated energies are very small. Leaving out the chemi-cal potential, which gives the same contribution for both states, the energy ofthe orbitally ordered ferromagnet is 〈H〉 = −0.0135W , and the energy of theantiferromagnetic state is 〈H〉 = −0.0085W . This is, compared to the chemi-cal potential, which gives contributions of 〈H〉 ≈ −W , a very small splitting.Looking at the different terms in the energy, the kinetic energy gives a largercontribution for the antiferromagnetic state than for the ferromagnetic state,


0 0.5 1 1.5 2U’/W

0

0.5

1

1.5

2

Occ

upat

ion

A siteB site

U’-J/2<0 U’-J/2>0

Figure 6.15: Occupation of neighboring sites in the charge ordered state for J/W = 2.The inter-orbital interaction becomes repulsive for U ′/W > 1. The lines are meant asguide to the eye.

-2 -1 0 1 2ω/w

0

1

2

3

ρ(ω

)W

U’/W=0.5U’/W=1.25U’/W=1.5

J/W=2

Figure 6.16: Spectral functions for the charge ordered state for U/W = 4, J/W = 2,and different U ′. The continuous (dashed) lines correspond to the occupied A-sites(nearly empty B-sites)

6.5 Summary 115

while the interaction terms increase the energy of the antiferromagnet. Varyingthe parameters J and U ′, I always find the antiferromagnetic state exhibitingthe higher energy.For large Hund’s coupling J/2 > U ′ the inter-orbital density-density interactionbecomes attractive. Although at a first glance such a large Hund’s coupling ap-pears unphysical, one can have situations, for example Jahn-Teller coupling tophonons, which can lead to additional contributions to J . In this case such anattractive interaction can effectively be generated and the physics will changedramatically. The first thing one notes is that it becomes very difficult tostabilize fillings other than n = 2 or n = 0. Inspired by this difficulty, I inves-tigated charge ordered phases in this parameter regime. And indeed for suffi-ciently large Hund’s coupling J it is possible to stabilize a charge ordered stateat quarter-filling, with alternating almost doubly occupied and nearly emptysites. The occupation of the different sites for this charge ordered phase can beseen in figure 6.15. The vertical line marks the point, where the inter-orbitaldensity-density interaction changes from attractive in the left part to repulsivein the right part. Remarkably, even for repulsive density-density interaction,one can stabilize such charge ordered states. Of course, the occupation of neigh-boring sites approach each other. The end of the curves denote the point, atwhich it was impossible to stabilize this phase. From the two-orbital Andersonmodel one knows that the Kondo temperature [52] decreases exponentially andthe doping dependence around quarter-filling becomes steeper for increasingHund’s coupling J . For an extended lattice this results in the charge orderedstate, even for repulsive interaction U ′−J/2 > 0. The spectral functions in thisphase can be seen in figure 6.16. This charge density wave is a perfect insulator,too. The electron number from neighboring sites add up to two, which meansthat there is an average of one electron per site. When increasing U ′, the gapshrinks. Interestingly, the spectral function of the double occupied site possessan additional peak at ω/W ≈ 5. As the broadening of the NRG is very largein this regions, one cannot deduce the structure of these peaks or tell what isthe origin of them.

6.5 Summary

This chapter dealt with the situation of two interacting degenerate bands. It isa basic model for an eg-band, as it can be found in transition metal oxides withcubic symmetry. Like the one-orbital Hubbard model, the half-filled systemis dominated by super exchange resulting in an antiferromagnetic phase. Iespecially looked at the interaction strength of U/W = 4. For this interactionthe system changes into an incommensurate spin density wave phase whendoping away from half-filling.In contrast to the one-orbital Hubbard model, the multi-orbital Hubbard modelpossesses a natural tendency towards ferromagnetism. For large enough Hund’scoupling J and density-density interactions U and U ′ the system forms a ho-mogeneous ferromagnetic state at 〈n〉 ≈ 1.5. Depending on the value of J thisphase can extend to occupations smaller than one.


Quarter-filling is a very special point for the two-orbital Hubbard model. Be-sides a homogeneous ferromagnetic metal, one can stabilize an orbitally orderedferromagnetic insulator for large U ′. Thus one can find a ferromagnetic metalinsulator transition at quarter-filling. First studies in the temperature depen-dence of the phases at quarter-filling seem to indicate that also a paramagneticmetal insulator transition can be found. Such metal insulator transitions accom-panied by a ferromagnetic phase transition provide the possibility to observesimilar effects as the colossal magneto resistance effect in manganites. There-fore, the dependence of this MIT on temperature and magnetic field should bestudied in more detail. Of course, in the manganites the just mentioned phe-nomenon occurs below quarter-filling and as a result of a transition between aferromagnetic metal and a paramagnetic insulator, but the ingredients for thiscan already be found in this very simplified model. Including the neglected t2g-spin and the coupling to the lattice, one can expect to describe the manganites.But as the strengths of the neglected effects depend on the exact manganitecompound, one can perhaps even hope that one can find a compound, whichnearly shows the effects analyzed in this chapter.Besides the just mentioned two ferromagnetic phases at quarter-filling, I addi-tionally observed an antiferromagnetic insulator and an charge ordered insula-tor. The charge ordered state exists only for very strong Hund’s coupling, whichfavors having two paired electrons at one site. In the same parameter region,in which the orbitally ordered ferromagnet exists, I found an antiferromagneticinsulator. But it seems as if this state has always the higher energy comparedto the ferromagnetic state.

CHAPTER 7

Summary and Outlook

This work dealt mainly with the topic of magnetic phases in the Hubbard model,which is a minimal model for analyzing the competition between localizationinduced by strong repulsive electron-electron interaction and delocalization in-duced by the electron hopping. This competition and the Pauli principle arethe basic ingredients for magnetism as being observed in transition metal ox-ides. Additionally, the orbital degeneracy of the d-orbitals plays an importantrole in these materials. For better understanding the electronic properties inthese materials, I have analyzed the influence of different hopping processes,the ratio between the interaction and the hopping, the filling of the lattice, andthe degeneracy of the orbitals. Thus, I tried to identify the influence of each ofthe mentioned properties. For the purpose of calculating magnetic phases forthe Bethe lattice I used the dynamical mean field theory (DMFT). Being exactin infinite dimensions it relates the lattice model to a quantum impurity model,which has to be calculated self-consistently.

DMRG as an Impurity Solver

Most of my results were gained using the numerical renormalization group(NRG), which is a well established method being able to calculate thermo-dynamic and spectral functions for a wide range of parameters. Additionally,I tested a newly developed code for the density matrix renormalization group(DMRG) as an impurity solver. The DMRG has the ability to perform groundstate calculations for impurity models allowing for the calculation of groundstate expectation values. For computing dynamical spectral functions, theDMRG uses an internal broadening, which results in a Lorentz convolutionof the spectral function. Compared to the NRG, this broadening is much largerfor frequencies near the Fermi energy, but smaller far away from it. Thus the

118 Summary and Outlook

DMRG is able to capture the behavior of the band edges and the substructuresin the Hubbard bands, but eventually fails to correctly resolve the Kondo peakor small gaps at the Fermi energy. Unfortunately, this results in problems whenusing the DMRG as impurity solver within the DMFT. It can be shown thatthe half-filled one-orbital Hubbard model forms an antiferromagnetic insulatorfor every finite repulsive value of the Hubbard interaction U > 0. This resultcan be reproduced by the NRG, but the DMRG fails if the interaction value istoo small, resulting in a very small gap, which cannot be resolved anymore.But the DMRG has a very big advantage to the NRG. For calculating dynamicalproperties properly within the NRG, one is limited to the one- or two-orbitalcase, because the local Fock space grows exponentially. Within the DMRG onecan further split up the local Fock space, resulting in a linear growth of thechain length when increasing the orbital number. This is supposed to enableone to perform calculations for multi-orbital Hubbard models, perhaps eventaking into account all 5 d-bands. Further studies on this subject should bedone.

One-Orbital Hubbard model

In my thesis I showed many results on magnetic phases for the one- and two-orbital Hubbard model. The magnetic phase diagram of the one-orbital Hub-bard model for a Bethe lattice with nearest neighbor hopping only is mainlycovered by an antiferromagnetic insulating phase exactly at half-filling and aparamagnetic metallic phase away from half-filling. However, for large inter-action strengths, there is an incommensurate spin density wave phase next tohalf-filling, which mainly stems from the antiferromagnetic fluctuation at half-filling. Interestingly, there is a transition from phase separation between theantiferromagnetic phase and the paramagnetic phase to this incommensuratephase exactly at the point, at which the paramagnetic metal insulator transitionwould occur. This stresses the connection between metal insulator transitionsand magnetic phases. This incommensurate phase manifests itself in the DMFTas a non-convergent solution. I proposed here a way how to stabilize such phasesfor a Bethe lattice, but unfortunately the method seems to be rather unstableand unreliable for finding such solutions. In order to better understand suchphases one should perform calculations for different lattices for which k-vectorscan be defined, so one can look directly for spin density wave states.Introducing a next nearest neighbor (NNN) hopping for the Bethe lattice changesthe density of states, which becomes asymmetric and develops a singularity.This imposes frustration to the Neel-state at half-filling. It was argued thatfrustration helps pushing the metal insulator transition out of the antiferro-magnetic dome. Increasing the frustration suppresses the Neel-temperature aswell as the critical end point of the paramagnetic metal insulator transition.Only for NNN-hopping comparable in strength to the NN-hopping the para-magnetic metal insulator transition reaches out of the magnetic phases whenincreasing the temperature. This is the scenario seen in V2O3, but one stillneeds very large values of NNN-hopping. Additionally, one can observe thatfor such large NNN-hopping the Neel-state is replaced at half-filling by a spin

119

density wave.Surprisingly, for comparable NN- and NNN-hopping one can still stabilize anantiferromagnetic Neel-state away from half-filling. Thus frustration changesthe filling at which a Neel-state is stable. Additionally, one can now find aferromagnetic state, which is due to the singularity in the free density of states.Ferromagnetic states cannot be found for a Bethe lattice with semi-ellipticdensity of states. First introducing a singularity into the density of statesmakes ferromagnetic order possible. The ferromagnetic state reaches to theparameter regions, where the antiferromagnetic state can be stabilized, too.Comparing the energies of the ferromagnetic and the antiferromagnetic statesin the fully frustrated model, there seems to be a phase transition between theferromagnetic and the antiferromagnetic state. But the latter state has nearlyequal energy to the paramagnetic state in this region. One could think of thepossibility that in this parameter region even more magnetic states are stabledue to the imposed frustration. But this is only a conjecture.

Two-Orbital Hubbard model

Finally, I showed results for the two-orbital Hubbard model describing corre-lations in a degenerate eg-band. Close to half-filling, the model shows similarphysics as the one-orbital Hubbard model. But away from half-filling, the two-orbital Hubbard model possesses even for symmetric free density of states thetendency towards ferromagnetism due to the degenerate orbitals. The mostinteresting point in the two-orbital Hubbard model is quarter-filling. There Icould observe a clash of four different long range ordered phases. There is aphase transition between two different ferromagnetic states. One is a homo-geneous metal, the other an orbitally ordered insulator. This phase transitioncan, for example, be triggered by changing the interaction parameters. Thus ametal insulator transition was found. Such transitions have always been verypopular in the research of strongly correlated materials. The metal insulatortransition observed here is strongly connected to the ferromagnetic phases ob-served at quarter-filling. It should be very interesting to study the influence oftemperature and magnetic fields on this phase transition in more detail. If thetransition point can be changed by applying a magnetic field, it would directlyconnect the resistivity to magnetic fields, which then can eventually be usedfor measuring magnetic fields. Very similar effects can be seen in manganitesin the form of the colossal magneto resistance effect.Besides these ferromagnetic states also antiferromagnetism and charge ordercould be found at quarter-filling. Charge order can especially be found for verylarge Hund’s coupling, which favors states where two electrons are paired onone site.This thesis explored some of the complexity, for which electronic correlations inan one- and two-orbital Hubbard model can be responsible. But mentioning thecolossal magneto resistance effect in manganites, one should remember that thecoupling to the very localized t2g-states and to the lattice is of great importance.Therefore, after studying the two-orbital Hubbard model, one should includethese degrees of freedom step by step. By this one can analyze the effects for

120 Summary and Outlook

which each coupling is responsible and finally build up a very powerful modelfor manganites and other transition metal oxides.

Appendices

APPENDIX A

Calculation of the Ground State

Energy

In two parts of this thesis I calculate the kinetic energy of the DMFT solution.Especially in the case of the frustrated lattice, the integrals are numericallydifficult, as singularities are involved. Therefore, parts of the integrals areperformed analytically improving the accuracy. I will now shortly describe howone can treat the occurring integrals. The kinetic energy for T = 0, see chapter5.4.6, can be written as

〈HT 〉N

=

∞∫

−∞

dθǫ(θ)ρ(θ)

0∫

−∞

dω

(

− 1

π

)

Im1

ω + µ− ǫ(θ) − Σ(ω + i0)

ǫ(θ) = t1θ + t2(θ2 − 1)

ρ(θ) =1

2π

√

4 − θ2.

As the self-energy is a calculated set of data points, the frequency integral hasto be performed numerically. But the θ-integral, can be performed exactly, inwhich the self-energy enters only as a complex parameter. The θ-integral canbe written as

2∫

−2

dθ(

t1θ + t2(θ2 − 1)

)

√4 − θ2

ζ − (t1θ + t2(θ2 − 1)),

where ζ = ω + µ − Σ(ω + i0). This integral can be solved by transformingit onto a complex contour by two consecutive substitutions. Firstly, writing

124 Calculation of the Ground State Energy

θ = 2cos(φ) gives

∫ 0

−πdφ2 sin(φ)

(

2t1 cos(φ) + t2(4 cos(φ)2 − 1)) 2 sin(θ)

ζ − (2t1 cos(θ + t2(4 cos(θ)2 − 1)))

Secondly, one transforms this integral to the closed complex unit circle by sub-stituting cos(φ) = 1

2 (exp(iφ)+exp(−iφ)) and sin = 12i (exp(iφ)−exp(−iφ)) and

using symmetry properties for performing the integral from∫ π−π. Simultane-

ously one can write z = exp(iφ) and 1/z = exp(−iφ). The integral thus can bewritten as

−1

2i

∮

S1

1

z

(

z − 1z

)2(

t1(

z + 1z

)

+ t2

(

(

z + 1z

)2 − 1))

ζ −(

t1(

z + 1z

)

+ t2

(

(

z + 1z

)2 − 1)) . (A.1)

Sorting powers of z, this reads

−1

2i

∮

S1

1

z3

(

z2 − 1)2 (

t1(z3 + z) + t2((z

2 + 1)2 − z2))

ζz2 − t1(z3 + z) − t2((z2 + 1)2 − z2).

The roots of the denominator polynomial can be found easily looking in equation(A.1). Thus, one is able to use the residue theorem to evaluate this integral fora given self-energy at frequency ω. The remaining ω-integration can then beperformed numerically.

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Thanks

Finally, I want to say “THANK YOU”.

First, I want to thank my adviser Thomas Pruschke. He has always an opendoor for questions. It was a pleasure and honor working, drinking coffee, andvisiting temples with him. I am very grateful for his support, trust, and friend-ship.I also want to thank Frithjof Anders for agreeing to review this thesis. Addi-tionally, it is always enlightening to discuss with him on conferences.

At this point I want to thank my family for their interest and support duringthe last years.

I thank all the people, who read parts of this thesis and gave me some goodadvice: Andreas, Katharina, Ludger, Oliver, Piet, Sebastian, Sigrun, and Till.Especially, I thank Andreas, who also has always an open door for questions.Additionally, I thank Ansgar, Anton, Piet, and Till for playing some good gamestable-soccer with me, and Riccardo for all the good Pizza and Coffee.

During my thesis, I was allowed to visit Kyoto for nearly three months. Forgiving me this opportunity, I give thanks to Kawakami-sensei, GCOE, andPruschke-sensei. I also want to thank Koyama-san and Noda-san for theirfriendship and support in Kyoto.

Lebenslauf

Personliche DatenName, Vorname Peters, RobertAnschrift Im Hassel 13, 37077 GottingenMail [email protected] DeutschGeburtstag 15. Dezember 1981Geburtsort Karl-Marx-Stadt (heute Chemnitz)

Werdegang1988-2000 Grundschule und Gymnasium in Chemnitz

Abschluss: Abitur 20002000-2001 Wehrpflicht in Cham und Regensburg

2001-2003 Grundstudium der Physik an der TU Chemnitz2003-2006 Studium der Physik an der Universitat Gottingen

Abschluss: Physikdiplom 20062006-2009 Promotion in Physik an der Universitat Gottingen

KenntnisseSprachen Englisch, Japanisch (Grundkenntnisse)

magnetic phases in the hubbard model

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